UC-NRLF 


$B    3Db    371 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/firstbookinarithOOcalirich 


CALIFORNIA 

STATE   SERIES   SCHOOL  TEXT-BOOKS 


FIRST   BOOK 


IN 


iVRITHlVEETIC 


COMPir.ED  BV  THE 
SXAXE  XEXX-BOOK  COMMITTEE 

AND  APPROVED  BY  THE 
STATE     BOARD    OF    EDUCATION 


SACRAMENTO: 

>UPE 

IS- 


W.    W.    SHANNON,    SUPERINTENDENT   STATE   PRINTING 


Copyright,  1905,  by 
THE  STATE  OF  CALIFORNIA 


Copyright,  1905,  by 

J.  W.   McCLYMONDS  and  D.   R.  JONES. 

Entered  at  Stationers'  Hall,  London. 

EDUOAXrON  -DEPT , 


In  the  compilation  of  this  work  certain  matter  from  McCly- 
tnondt  anil  Jones's  Blementary  Arithmetic  has  been  VM«d.  All 
•wcA  matter  is  protected  by  the  copyright  entry  nottd  abovt. 


6e— 50M— 2,  '10 


PREFACE 

It  is  quite  generally  admitted  that  the  results  obtained 
in  the  field  of  primary  arithmetic  are  by  no  means  com- 
mensurate with  the  attention  which  the  subject  receives 
in  our  schools.  After  several  years  of  earnest  effort,  the 
pupils  too  often  show  little  or  no  insight  into  number 
relations.  Furthermore,  the  ability  to  handle  numbers 
with  accuracy  and  a  fair  degree  of  facility  is  also  wanting. 
This  lack  of  results,  we  believe,  is  due  chiefly  to  the  inade- 
quacy of  the  methods  commonly  pursued  in  the  teaching 
of  this  subject.  It  too  frequently  occurs  that  beginners 
are  introduced  to  meaningless  abstractions  (which  are 
retained  only  with  difficulty,  notwithstanding  their  fre- 
quent repetition),  and  are  hurried  on  to  the  fundamental 
operations,  which  are  rarely  well  mastered. 
•  The  work  outlined  in  Chapter  I  of  this  book  is  designed 
to  prepare  the  pupils  for  the  intelligent  mastery  of  the 
fundamental  operations  as  presented  in  the  succeeding 
chapters.  Through  the  application  of  number  to  objects, 
an  insight  into  number  relations  and  the  common  opera- 
tions is  gained.  Throughout  this  chapter  the  memorizing 
of  facts  is  subordinate  to  the  getting  of  ideas. 

The  mastery  of  the  fundamental  operations  is  taken  up 
in  the  succeeding  chapters,  and  a  well-developed  method 
is  provided  for  each  operation.  The  general  plan  of  work 
is  a  simple  one.     The  pupils  are  required  to  memorize  a 

M187502 


4  PREFACE 

few  number  facts  and  to  apply  these  until  they  have  be- 
come perfect  reflexes,  before  new  facts  are  introduced. 

A  constant  review  is  provided,  as  the  facts  of  previous 
lessons  are  involved  in  the  drill  exercises  which  follow  the 
several  lessons. 

Simple  work  in  fractions  and  denominate  numbers  is 
introduced  in  the  first  lessons,  and  continued  throughout 
the  book.  A  balance  is  maintained  between  the  mechani- 
cal work  on  the  one  hand  and  the  solution  of  problems  on 
the  other.  The  problems  are  of  a  practical  character,  and 
are  drawn  largely  from  the  field  of  everyday  experience. 
Whenever  a  problem  of  a  given  type  seems  to  offer  a  lan- 
guage difficulty,  several  such  problems  are  given  in  suc- 
cession. Such  repetition  frequently  occurs  in  the  earlier 
pages  of  the  text. 

Frequent  requests  for  detailed  information  with  refer- 
ence to  the  methods  herein  contained,  and  favorable  reports 
from  teachers  and  superintendents  who  have  found  these 
methods  helpful  in  their  work,  have  led  to  the  preparation 
of  this  text,  which,  we  trust,  will  be  found  of  service  in  its 
field  of  intended  usefulness. 


CONTENTS 

CHAPTER  I 
Preliminary  Type  Lessons 

PAGE 

Simple  Directions  —  Magnitude  —  Counting  —  Comparison  of 
Quantities  —  Grouping  —  Writing  Numbers  —  Division  by 
Measurement  and  Partition  —  Measurements — Forms  — 
Fractions  —  Difference  —  Number  Stories  —  Measure  of 
Time  —  Summary 7-33 

CHAPTER   II 

General  Introduction 

Steps  in  Addition,  Subtraction,  Multiplication,  and  Division       34-42 

CHAPTER  III 

Addition  and  Subtraction 

Notation  —  Numeration  —  Addition  —  Subtraction  —  Objective 
Fractions  —  Compound  Numbers  —  Multiplication  —  Divi- 
sion         43-114 

CHAPTER  IV 

Multiplication  and  Division 

Addition  —  Subtraction  —  Multiplication  —  Division  —  Simple 

Fractions  —  Compound  Numbers 115-187 

CHAPTER  V 

Common  and  Decimal  Fractions 

Written  Fractions  —  Decimals  —  Compound  Numbers  —  Per- 
centage—  Interest 188-244 


6  CONTENTS 

CHAPTER  VI 
Denominate  Numbers 

'     PAGE 

Tables  —  Time  Problems  —  Measure  of  Length — Square  and 
Cubic  Measure  —  Lumber  Measure  —  Cash  Account  — 
Angles  —  Roman  Notation 245-256 


ELEMENTARY  ARITHMETIC 

CHAPTER  I 
PRELIMINARY  LESSONS 

To  THE  Teacher.  The  lessons  of  this  chapter  are 
designed  to  indicate  the  nature  of  the  work  that  should  be 
done  by  the  class  before  the  text  is  placed  in  the  hands  of 
the  pupils.  Space  does  not  permit  of  the  introduction  of 
sufficient  material  to  furnish  all  of  the  exercises  which  the 
pupils  will  need.  The  lessons  in  this  chapter  should,  there- 
fore, be  regarded  as  type  lessons  which,  the  teacher  is  to 
expand  and  to  supplement  to  meet  the  needs  of  the  class. 
It  is  not  expected  that  all  of  the  work  suggested  in  any 
one  of  the  lessons  will  be  given  at  any  one  time.  Several 
parallel  lines  of  work  are  suggested  in  the  various  lessons, 
and  these  should  be  carried  on  together,  the  work  gradu- 
ally increasing  in  difficulty  until  the  pupils  have  finally 
mastered  all  of  the  work  indicated  in  each  of  the  lessons. 

No  abstract  number  work,  aside  from  counting  and  the 
reading  and  writing  of  numbers,  is  provided,  and  none 
should  be  given.  The  pupils  should  deal  with  number  in 
its  relation  to  things,  and  not  with  abstract  number  facts. 
In  the  exercises  suggested  in  this  chapter,  the  pupils  them- 
selves play  an  important  part.  They  are  required  to  c?o, 
as  well  as  attend  to  what  is  done  by  others.  They  are  led 
to  discover  number  relations  in  the  quantities  that  are 

7 


g  "   *  *'*  PREJJMINARY  LESSONS 

handled  by  them,  and  to  express  these  reiations  in  correct 
language. 

These  lessons  are  presented  in  the  form  of  questions 
and  directions  given  by  the  teachers,  as  serving  best  to 
illustrate  the  methods  to  be  followed  in  presenting  them 
to  the  class.  No  exercise  should  be  continued  so  long 
that  the  pupils  will  begin  to  lose  interest  in  it.  The 
teacher  is  expected  to  use  whatever  objects  may  be  at 
hand,  and  to  vary  the  objects  frequently.  A  box  of  1-inch 
cubes  will  be  found  very  useful  in  this  as  well  as  in  subse- 
quent number  work.  They  are  easily  handled,  and  may 
be  conveniently  arranged  to  show  relative  quantities,  etc. 

The  pupils  should  be  encouraged  to  express  themselves 
freely,  but  at  the  same  time  correctly.  Sufficient  time 
should  be  taken  to  acquaint  the  pupils  with  the  language 
forms  involved  in  these  lessons.  If  properly  presented, 
the  work  suggested  in  this  chapter  should  not  only  give 
tlie  pupils  familiarity  with  simple  number  facts,  and  an 
insight  into  the  common  operations  with  number,  but 
should  also  establish  habits  that  will  be  found  extremely 
valuable  in  all  subsequent  number  work.  Some  of  the 
number  facts  developed  in  these  lessons  are  designated  as 
facts  to  be  learned.    These  should  be  perfectly  memorized. 

LESSON   I  — MAGNITUDE 

The  primary  purpose  of  Lesson  I  is  to  train  the  pupils 
to  hear  and  interpret  simple  directions.  The  secondary 
purpose  is  to  acquaint  the  pupil  with  the  language  forms 
used  to  denote  relative  position,  direction,  magnitude,  etc. 
After  giving  a  direction,  allow  sufficient  time  for  all  the 
pupils  to  interpret  the  direction,  and  to  image  its  execution. 
Should  the  pupil  called  upon  fail  to  execute  correctly  the 


MAGNITUDE  9 

direction  given,  do  not  call  upon  a  second  pupil  for  it, 
but  give  a  new  direction.  This  will  be  found  more  effec- 
tive in  keeping  the  attention  of  the  whole  class  than  the 
more  common  procedure,  namely,  that  of  permitting  the 
brighter  pupils  to  do  most  of  the  work.  Later,  return 
to  the  direction  upon  which  a  failure  was  made,  and  call 
upon  some  other  child  to  carry  it  out,  unless  you  have 
reason  to  think  that  the  pupil  who  once  failed  can  now 
execute  it  correctly. 

Simple  directions  are  to  be  executed,  showing  the  mean- 
ing of  the  following  and  similar  terms  :  in  your  right  hand^ 
in  your  left  hand,  to  the  right  of^  to  the  left  of  above,  below, 
nearer^  farther  from,  beside,  between,  in  front  of,  larger  than, 
taller  tha7i,  tallest,  shorter  than,  shortest,  smallest,  twice  as 
lo7ig  as,  one  half  as  long  as,  ttvice  as  far  from,  one  half  as 
far  from,  the  same  distance  from,  a  line^  at  the  end  of  in 
the  middle  of  etc. 

Illustration.  Have  two  boys  from  the  class,  say 
John  and  James,  stand  in  line  in  front  of  the  class,  John 
several  feet  to  the  right  of  James.  Problems:  1.  I  want 
some  one  to  stand  in  line  with  John  and  James.  2.  Stand 
in  line  with  the  two  boys,  to  the  right  of  John.  3,  Stand 
in  line^  to  the  left  of  James.  4.  Stand  in  line,  so  John  is 
to  your  right  and  James  is  to  your  left.  5.  Stand  in  line, 
the  same  distance  from  the  boys.  6.  Stand  in  line,  nearer 
John  than  James.  7.  Stand  in  line,  farther  from  James 
than  from  John.  8.  Stand  in  line,  one  half  as  far  from 
John  as  from  James.  9.  Stand  in  line,  to  the  right  of  John, 
one  half  as  far  from  John  as  from  James.  10.  Stand  out 
of  line,  but  the  same  distance  from  each  of  the  two  boys. 

Illustrati(3N.  With  colored  crayon  draw  on  the 
board  lines  of  different  lengths,  for  comparison.  Problems: 
1.    Alice,  tell  me  a  story  about  the  yellow  line  and  the 


10  PRELIMINARY  LESSONS 

blue  line.  Story  :  The  yellow  line  is  longer  than  the  blue 
line.  2.  Walter,  tell  me  a  story  about  the  red  line  and 
the  blue  line.  Story :  The  red  line  is  shorter  than  the  blue 
line.  3.  Mary,  tell  me  a  story  about  the  green  line  and 
the  red  line.  Story :  The  green  line  is  as  long  as  the  red 
line.  4.  Who  can  tell  me  a  story  about  the  orange  line 
and  the  blue  line  ?  Story :  The  orange  line  is  one  half  as 
long  as  the  blue  line.  5.  Who  can  tell  me  a  story  about 
the  yellow  line  and  the  red  line  ?  Story :  The  yellow 
line  is  two  times  (or  twice')  as  long  as  the  red  line. 
6.  Ethel,  tell  me  a  story  about  the  red  line,  the  blue  line, 
and  the  yellow  line.  Story:  The  red  line  and  the  blue 
line  together  are  as  long  as  the  yellow  line. 

Substitute  lines  drawn  with  white  crayon  and  lettered 
a,  ^,  <?,  etc.,  in  place  of  the  colored  lines,  and  treat  in  a 
similar  manner.  Have  the  pupils  point  to  the  lines  as 
they  name  them  in  giving  their  answers.  Later,  make 
written  statements  regarding  these  lines,  leaving  blanks 

for  the  pupils  to  fill  in,  thus  :   The  red  line  is than 

the  blue  line.     The line  is  shorter  than  the  red  line. 

Compare  the  length  of  the  board  with  its  width,  the 
height  of  the  room  with  its  width  and  length,  etc. 

LESSON   II  — COUNTING 

In  the  preparation  of  this  lesson,  it  is  presumed  that 
the  pupils  have  acquired  some  number  knowledge  before 
entering  school.  The  teacher  should  ascertain  as  soon 
as  possible  what  knowledge  the  pupils  have,  and  should 
adapt  the  lessons  to  the  needs  of  the  class. 

Illustration.  Problems :  1.  Harry,  bring  me  one 
book  from  the  table.  2.  Jane,  bring  me  two  books. 
3.    How  can  I  show  on  the  board  the  number  of  books 


COUNTING  11 

Jane  brought  me  ?  4.  Fred,  will  you  draw  a  line  on  the 
board  for  each  book  Jane  brought  me  ?  5.  Walter,  bring 
me  three  books.  6.  Lottie,  draw  lines  on  the  board  to 
show  how  many  books  Walter  brought  me.  7.  I  will 
write  on  the  board  the  figure  that  tells  how  many  lines 
you  made.  8.  I  can  show  the  number  of  books  Walter 
brought  me  in  two  ways :  1  1  1  books,  and  3  books. 
9.  Count  the  books,  one,  two,  three.  10.  Alice  may 
hold  three  books  in  her  hand.  11.  Mary  may  take  two 
of  the  books  that  Alice  has  and  place  them  on  the  desk. 
12.  How  many  books  have  you  now,  Alice  ?  13.  Lottie, 
hold  up  one  book.  14.  Hold  up  two  books.  15.  Hold 
up  three  books.  16.  I  want  some  one  to  build  a  pile  of 
three  books  and  a  pile  of  two  books.  17.  David,  point 
to  the  pile  of  three  books.  18.  Emma,  show  me  two 
girls.  19.  George,  stand  three  boys  in  line  on  the  floor. 
20.  Send  one  of  the  boys  to  his  seat.  21.  How  many 
boys  are  left  standing  ?  22.  I  will  show  on  the  board 
the  number  of  boys  we  have  been  talking  about: 

•         •         • 
12         3 

23.  Mary,  point  to  something  that  tells  three.  24.  Fred, 
point  to  something  else  that  tells  three. 

Vary  the  objects  counted,  and  continue  the  group 
pictures  through  twelve.  The  group  pictures  are  used 
to  represent  the  objects  which  the  pupils  have  handled. 
Disks  of  colored  paper  mounted  on  cardboard  may  be 
used  as  flash  cards  for  the  recognition  of  these  groups. 
The  disks  should  represent  any  object  whatever,  and  each 
disk  represent  whatever  unit  is  selected.  The  group 
three  may  represent  three  boys  or  three  apples,  or  it  may 


12 


PRELIMINARY  LESSONS 


represent  three  ones,  three  nickels,  or  three  tens.     The 
groups  through  twelve  ma}^  be  arranged  thus: 


•. 

• 
• 

• 

•  • 

•  • 

1 

2 

3 

4 

•         • 

•  •    • 

•  •    • 

••      •• 

•  •    • 

•  •    •  • 

•  •    •  • 

5 

6 

7 

8 

•  •    •  • 

•  •    •  • 

• 

•  •  o    •  •  • 
• • •     • •• 

10 


11 


12 


These  groups  should  always  represent  concrete  numbers, 
—  apples,  books,  blocks,  cents,  dimes,  etc. 

Continue  oral  counting  until  the  pupils  are  familiar 
with  the  number  scale.  Never  ask  a  pupil  to  "  count 
backwards."  The  attention  of  the  pupils  should  be  called 
to  the  place  that  a  given  number  occupies  in  the  number 
scale  by  such  questions  as  the  following ;  In  counting, 
what  number  comes  just  before  20?  What  number  comes 
next  after  17  ?  Twenty-seven  comes  before  what  number? 
From  the  exercises  in  counting  and  in  the  writing  of  the 
number  table,  the  pupils  should  know  wliat  number  is  one 
more  than  any  given  number,  and  what  number  is  one  less 
than  any  given  number. . 

Give  exercises  in  counting  by  2's  to  12,  using  objects. 
Count  by  2's  the  hands,  feet,  eyes,  etc.,  of  the  pupils  in 
the  class.  Count  by  2's  from  10  to  22.  Count  by  2*s 
from  20  to  32,  etc. 

Count  by  5's  to  15,  using  objects.     Count  by  5's,  using 


COMPARISON   OF   QUANTITIES 


13 


nickels.  Count  by  5\s  from  10  to  25,  from  20  to  35,  etc. 
Count  by  5's  the  minutes  on  the  clock  face. 

Count  by  lO's  to  110.  Count  by  lO's  from  any  given 
number,  such  as  1,  2,  5,  7,  etc.  to  110.  Use  the  number 
table  in  connection  with  this  work. 

Before  taking  up  the  study  of  the  text,  the  pupils  should 
be  able  to  count  serially  in  any  part  of  the  number  scale 
below  1000.  They  should  be  able  to  count  by  2's,  begin- 
ning with  even  numbers,  and  by  5's,  beginning  with  num- 
bers ending  in  5  and  0,  up  to  100.  They  should  be  able 
to  count  by  lO's  to  110,  beginning  at  any  place  in  the  first 
hundred  of  the  number  scale.  They  should  know  tlie 
numbsr  of  tens  there  are  in  30,  40,  80,  etc.  From  a  study 
of  the  number  scale,  they  should  know  what  one  more  than 
any  number  is,  and  what  the  sum  of  10  and  any  number  less 
than  10  is,  what  the  sum  of  20  and  any  number  less  than 
10  is,  what  the  sum  of  30  and  any  number  less  than  10  is, 
etc.     This  knowledge  is  essential  to  the  work  in  addition. 


LESSON   III  — COMPARISON   OF   QUANTITIES 

Using  the  inch  cubes,  have  the  pupils  build  "towers" 
with  as  many  blocks  as  the  figures  on  the  table  or  the 
board  specify,  thus : 


,r"-~Pn 


ai_ 


Illustration.     Problems:    1.    Fred,  tell  me  a  story 
about  the  tower  of  3  blocks  and  the  tower  of  2  blocks. 


14  PRELIMINARY  LP:SSONS 

Story :  The  tower  of  3  blocks  is  higher  than  the  tower  of 
2  blocks.  2.  How  many  towers  of  2  blocks  will  it  take 
to  make  a  tower  of  4  blocks?  3.  What  must  you  put 
with  a  tower  of  5  blocks  to  make  a  tower  of  7  blocks? 
4.  How  many  towers  of  2  blocks  can  you  make  out  of  a 
tower  of  6  blocks  ?  5.  What  must  you  do  with  a  tower 
of  6  blocks  to  make  a  tower  of  4  blocks  ?  Of  5  blocks  ? 
'Of  8  blocks?  6.  Find  a  tower  one  half  as  high  as  a 
tower  of  8  blocks. 

Using  blocks,  build  trains  of  as  many  cars  as  the  figures 
on  the  tracks  designate,  thus  : 

if^  ■ 


r  r  r  a 


r^rm 


.r  r  r  r  rx 


1.  How  much  longer  is  the  train  of  3  cars  than  the 
train  of  2  cars  ?  2.  What  will  you  have  to  put  with  the 
train  of  4  cars  to  make  a  train  of  6  cars  ?  3.  What  will 
you  have  to  do  with  a  train  of  5  cars  to  make  it  a  train  of 
3  cars  ?  4.  How  many  trains  of  two  cars  can  you  make 
out  of  a  train  of  8  cars?  5.  How  many  trains  of  3  cars 
must  we  put  together  to  make  a  train  of  6  cars?  Of 
9  cars  ?  6.  Break  the  train  of  10  cars  into  halves.  How 
many  cars  are  there  in  one  half  of  it?  How  many  are 
there  in  the  other  half  of  it  ?  7.  Join  a  train  of  2  cars  to 
a  train  of  3  cars.  It  is  as  long  as  a  train  of  —  cars. 
8.  Join  a  train  of  2  cars  to  a  train  of  6  cars.  It  is  as 
long  as  a  train  of  4  cars  and  a  train  of  —  cars.  9.  A 
train  of  6  cars  is  —  cars  longer  than  a  train  of  3  cars. 


GROUPS  15 

LESSON   IV  — GROUPS 

Separation  into  Groups.  Illustration.  The  teacher 
places  3  books  on  the  table.  Problems :  1.  Harry,  here 
are  some  books  on  the  table.  Take  up  some  of  them  in 
your  right  hand,  and  the  rest  in  your  left  hand.  2.  How 
many  books  has  Harry  in  his  left  hand?  3.  How  many 
has  he  in  his  right  hand  ?  4.  Name  the  number  in  each 
hand  as  he  holds  them  up.  5.  Count  them  as  he  puts 
them  together.  6.  Show  on  the  board  the  number  of 
books  Harry  holds  in  his  right  hand.  7.  Show  the  num- 
ber he  holds  in  his  left  hand.  8.  Harry,  place  the  books 
on  the  table.     9.    Annie,  divide  them  into  two  groups. 

Illustration.  Problems:  1.  "Fred,  place  4  books  on 
the  table.  2.  Separate  them  into  two  equal  piles.  3. 
Mary,  hold  up  one  half  of  the  books.  4.  Has  she  the  cor- 
rect number  ?  5.  How  many  books  is  Mary  holding  up  ? 
6.  How  many  groups  of  two  books  are  on  the  desk  ?  7.  How 
many  groups  of  two  books  is  Mary  holding?  8.  Mary, 
hold  the  other  group  of  two  books  in  your  left  hand.  9. 
How  many  groups  of  two  books  is  Mary  holding  ?  10.  How 
many  groups  of  two  books  is  she  holding  in  each  hand  ? 
11.  Mary,  place  the  books  together.  How  many  books 
have  you  altogether  ?  12.  Separate  the  groups  of  two 
books.  13.  How  many  groups  of  two  books  are  there  in 
four  books?  14.  Place  the  books  on  the  desk  in  twos. 
15.  Willie,  hold  up  one  half  of  the  books.  16.  How 
many  books  is  Willie  holding  up  ?  17.  Hold  up  all  of  the 
books.  18.  Place  two  of  the  books  on  the  table.  Two 
books  are  one  half  of  how  many  books  ? 

Illustration.  Problems:  1.  Place  four  books  in  a 
pile.  2.  Remove  the  books,  one  book  at  a  time.  3.  Re- 
move the  books,  one  2-book  group  at  a  time.     4.    Remove 


16  PRELIMINARY  LESSONS 

the  books,  one  book  and  a  3-book  group  at  a  time.  5.  Re- 
move the  books,  one  3-book  group  and  one  book  at  a  time. 
6.  Count  the  books  back,  one  book  (placing  down  the 
one  book),  four  books  (placing  down  the  3-book  group); 
two  books  (placing  down  the  2-book  group),  four  books 
(placing  down  the  other  2-book  group),  etc.  7.  Close 
your  eyes.  Count  the  books  by  ones  as  you  hear  them 
touch  the  table.  8.  Close  your  eyes.  Count  the  books 
by  twos  as  you  hear  them  touch  the  table.  9.  Close  your 
eyes  and  count  them  as  you  hear  the  3-book  group,  then 
the  one  book  touch  the  table.  10.  Close  your  eyes.  Count 
them  as  you  hear  the  one  book,  then  the  3-book  group 
touch  the  table.  11.  Close  your  eyes.  I  have  4  books 
in  my  hands.  In  one  hftnd  I  have  2  books.  In  the  other 
I  have  —  books. 

Grouping.  Illustration.  The  teacher  shows  two 
books  in  her  right  hand  and  one  book  in  her  left  hand. 
She  brings  the  two  groups  together  to  form  a  single  group. 
Problems:  1.  I  want  some  one  to  show  on  the  board, 
with  lines  or  figures,  the  number  of  books  I  hold  in  my 
right  hand.  2.  Mabel  may  show  the  number  I  hold  in  my 
left  hand.  3.  Martha  may  show  the  number  I  hold  in 
both  hands.  4.  Close  your  eyes.  Tliink  of  how  the  books 
looked  which  I  held  in  my  hands.  When  I  say  "  up,''  I  am 
going  to  hold  up  the  books  which  I  have  in  my  right  hand. 
5.  Up.  Fred,  you  may  tell  me,  without  looking,  how 
many  books  I  am  holding  up.  Think  of  how  the  books 
in  my  hands  looked  as  I  put  them  together.  6.  I  am  go- 
ing to  bring  them  together.  When  you  hear  them  touch, 
tell  me  how  many  books  there  are  in  the  group.  Using 
two  objects  and  two  objects,  three  objects  and  one  object, 
etc.,  continue  this  lesson  to  twelve  as  a  sum.  Do  not  aim 
to  have  the  pupils  memorize  the  answers. 


WRITING  NUxMBERS 


17 


LESSON   V  — WRITING   NUMBERS 

Figures.  Tlie  teacher  should  exercise  much  care  in  the 
making  of  figures,  particularly  those  given  as  models  to 
the  class.  They  should  he  free  from  all  unnecessary  loops, 
and  should  all  be  of  the  same  height,  that  proper  align- 
ment may  be  had  in  writing  columns  of  figures.  The 
pupils  should  be  taught  how  to  make  their  figures  prop- 
erly, and  to  write  the  numbers  in  a  horizontal  or  a  vertical 
line,  preferably  the  latter.  The  arrangement  shown  in  the 
number  table  will  be  found  helpful. 

NUMBER   TABLE 


1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

55 

65 

75 

85 

95 

G 

16 

26 

36 

46 

56 

66 

76 

86 

96 

7 

17 

27 

37 

47 

51 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

0 

20 

30 

40 

50 

60 

70 

80 

90 

100 

Writing  Numbers.  Illustration.  Problems:  1.  Show 
with  your  counters  two  ones  (11);  three  ones  (111); 
four  ones  (1111);  ten  ones  (1111111111).  2.  Show 
with  your  counters  one  ten.  3.  Show  one  ten  and  two 
ones.  4.  Begin  with  1  and  write  in  a  column  the  figures 
that  stand  for  the  numbers  through  10.  5.  In  the  num- 
ber 10,  which  figure  stands  for  no  ones?  This  figure  is 
read  naught  (cipher,  or  zero).  6.  The  ones'  place  is  the 
first  or  right-hand  place.  What  is  the  greatest  number 
1st  T?k  AniTii— 2 


18  PRELIMINARY   LESSONS 

that  can  be  written  in  units'  or  ones'  place  ?  7.  The 
next  place  to  the  left  is  called  tens'  place.  It  tells  the 
number  of  tens.  8.  The  units'  place  tells  the  number  of 
ones.  9.  In  10  which  figure  tells  how  many  tens  there 
are?  10.  What  does  the  naught  tell?  11.  Show  with 
your  counters  12,  using  your  bundle  of  ten.  12.  Show 
with  your  counters  another  ten.  13.  Write  in  a  column 
to  the  right  of  the  first  column  the  numbers  in  the  sec- 
ond ten.  14.  Show  with  your  counters  two  tens  and 
two  ones.  15.  Show  with  your  counters  27.  Continue 
this  work  until  the  number  scale  has  been  developed 
to  100.  The  tens  in  the  number  table  are  arranged  to 
correspond  to  the  tens  in  the  number  scale.  The  pupils 
should  know  that  two  tens  are  20,  three  tens  are  30,  etc. 
The  number  table  should  be  used  as  the  basis  of  count- 
ing by  tens.  Give  much  drill  in  reading  and  writing 
numbers  to  120.  Read  101  one  hundred  one,  and  not  one 
hundred  and  one.  Give  many  problems  like  the  fol- 
lowing :  1.  If  121  is  read  one  hundred  twenty-one,  how 
do  you  read  122?  126?  128?  2.  How  do  you  write  one 
hundred  twenty-five  ?  3.  Change  one  figure  in  125  and 
make  it  read  one  hundred  thirty-five.  4.  In  234  what  does 
the  2  tell  ?    What  does  the  3  tell  ?    What  does  the  4  tell  ? 

5.  How   many   hundreds    are    there    in   one   thousand  ? 

6.  Count  by  lOO's  to  one  thousand.  7.  One  thousand  is 
written  thus  :  1,000.  8.  Two  thousand  is  written  thus  : 
2,000.  9.  We  write  one  thousand  two  hundred  thirty- 
four  thus  :  1,234.  10.  For  our  convenience  in  reading 
the  numbers  we  shall  always  separate  the  part  of  the  num- 
ber that  tells  the  thousands  from  the  part  of  the  number 
to  the  right  of  it.  11.  How  many  places  are  there  to  the 
right  of  the  comma  in  1,234  ?  12.  There  must  always  be 
three  places  to  the  right  of  the  comma.     13.    These  three 


WRITING  NUMBERS  19 

places  constitute  the  first  period,  or  ones'  period.  14.  How 
many  ones  are  there  in  234  ?  Read  234  two  hundred 
thirty-four.  Do  not  give  the  name  of  the  period,  which 
is  ones'.     15.    One  thousand  one  is  written  1,001. 

Give  special'  attention  to  the  reading  and  writing  of 
numbers  between  100  and  120,  and  between  1,000  and 
1,100.  It  will  be  helpful  at  first  to  separate  the  periods 
by  a  vertical  line  instead  of  a  comma,  as  it  will  emphasize 
the  fact  that  each  period  must  be  taken  separately.  The 
number  350,406  consists  of  two  parts,  either  of  which  can 
be  read  without  much  difficulty.  The  part  at  the  left 
of  the  line  (or  comma)  should  be  read  without  reference 
to  the  rest,  and  its  name  (thousands)  given.  The  part  at 
the  right  of  the  line  (or  comma)  should  be  read  as  though 
it  stood  alone.  At  the  close  of  this  work  the  pupils 
should  be  able  to  read  and  write  correctly  numbers  of 
two  periods,  without  any  hesitation. 

Rapid  Drill  Exercises.  With  the  class  at  the  board, 
give  frequent  exercises  in  writing  numbers.  Illustra- 
tion. Directions:  Hold  the  crayon  in  your  right  hand 
and  the  eraser  in  your  left  hand.  Attend  carefully  to 
what  is  said,  and  do  only  what  you  are  told  to  do. 
Exercises:  Write  20.  Change  one  figure  and  make  it 
read  30.  Change  one  figure  and  make  it  read  40 ;  50 ; 
60 ;  90.  Change  one  figure  and  make  it  read  19 ;  13 ; 
15  ;  11 ;  71 ;  21 ;  91 ;  95.  Erase  one  figure  and  make 
it  read  5.  Add  one  figure  and  make  it  read  ^^.  Add 
another  figure  and  make  it  read  465.  Change  one  figure 
and  make  it  read  405 ;  105 ;  175 ;  etc.  Write  25.  Add 
one  figure  so  as  not  to  change  the  reading  (025).  Add 
one  figure  and  make  it  read  1,025.  The  value  of  this 
exercise  will  depend  largely  upon  the  teacher's  ability 
to  conduct  it  with  dispatch. 


20  PUELIMIXAIIY  LESSONS 

LESSON  VI  — DIVISION  BY  MEASUREMENT 

Division  by  Measurement.  The  teacher  should  not 
confuse  division  by  measurement  with  division  by  parti- 
tion. 

Illustration.  Place  six  counters  or  blocks  in  a  pile 
near  one  end  of  the  table.  Near  the  other  end  place  two 
similar  counters,  to  be  used  as  a  measure  in  measuring 
this  group.  Problems :  1.  Measure  this  pile  of  counters, 
using  the  measure  which  you  see  at  this  end  of  the  table. 
Arrange  your  counters  at  the  center  of  the  table. 

&        &        &  •        0 

g     g     g  & 

2.  How  many  times  did  you  use  the  measure  ?  3.  How 
many  two  counters  are  there  in  six  counters?  4.  Mary 
may  measure  the  same  pile  of  counters,  using  the  same 
measure.  5.  Measure  the  same  pile  with  three  counters 
as  a  measure.  6.  How  many  three  counters  are  there  in 
six  counters  ?  7.  Measure  the  same  pile  with  one  coun- 
ter as  a  measure.  8.  How  many  times  one  counter  are 
there  in  six  counters  ? 

Continue  this  work  until  you  have  used  one,  two,  three, 
four,  five,  and  six  counters  as  units  of  measure  in  measur- 
ing piles  of  not  more  than  twelve  objects. 

Measure  oft*  a  space,  say  six  feet,  on  the  board  as  the 
distance  to  be  measured.  Use  the  yard  stick  as  the  unit 
of  measure.  Use  the  foot  rule  as  the  unit  of  measure. 
Use  also  a  6-inch  rule  as  a  unit  of  measure.  The  purpose 
of  this  exercise  is  not  to  find  the  distance  between  the  two 


mCHES,   FEET,   AND   YARDS  21 

points,  but  to  develop  the  idea  of  units  of  measure.  Prob- 
lems:  1.  How  many  times  did  you  apply  the  longest 
measure  ?  2.  How  many  times  did  you  apply  the  short- 
est measure  ?  3.  How  many  times  did  you  apply  the 
other  measure  ?  4.  Which  did  you  apply  the  greater 
number  of  times,  the  longest  measure  or  the  shortest 
measure  ? 

Use  the  pint,  quart,  and  gallon  as  units  of  measure. 

Have  the  pupils  find  how  many  different  units  of  meas- 
ure can  be  applied  to  a  pile  of  twelve  counters ;  to  a  pile 
of  eight,  six,  nine,  seven,  five,  and  eleven  counters.  Have 
them  select  the  number  of  counters  as  high  as  twelve,  that 
can  be  measured  with  two  counters  without  a  remainder. 
Associate  this  list  with  counting  by  2's.  Even  and  odd 
numbers  will  be  recognized  by  this  test,  and  the  pupil 
should  become  familiar  with  them  through  this  experience. 

Use  the  flash  cards  for  recognition  of  units.  The  group 
for  five  is  two  2's  and  1,  five  I's,  or  one  5  ;  the  group  for 
six  is  two  3's,  three  2's,  six  I's,  one  6,  one  4  and  one  2. 
The  pupil  may  not  be  able  on  account  of  the  arrangement 
of  disks  to  see  the  group  5  and  1.  No  attempt  should  be 
made  to  have  the  pupils  memorize  these  facts. 

LESSON   VII  — INCHES,   FEET,    AND   YARDS 

Foot  and  Yard.  Illustration.  Draw  on  the  board 
two  vertical  lines  nine  feet  apart,  or  such  distance  apart 
as  can  be  measured  with  the  foot  rule  and  the  yard  stick 
without  a  remainder.  Letter  the  lines  a  and  h.  Problems : 
L  How  far  is  it  from  line  a  to  line  6,  Frank?  2.  How 
far  do  you  think  it  is,  Henry  ?  3.  Alice,  how  can  we  find 
out  liow  far  it  is  ?  4.  Shall  I  measure  it  with  a  string  ? 
5.    Tell   me  of  some   measure   I   can   use   to   measure  it. 


22  PRELIMINARY  LESSONS 

6.  Does  any  one  know  of  any  other  measure  that  can  be 
used  ?  7.  Fred  and  Walter  may  measure  it  with  the  yard 
stick.  Martha  may  write  their  answer  on  the  board. 
8.  Are  they  measuring  it  correctly,  class?  9.  Henry 
and  John  may  measure  it  with  the  foot  rule.  Lottie 
may  write  their  answer  on  the  board.  10.  With  which 
measure  do  you  think  it  is  easier  to  measure  it,  Ethel? 
11.  Why  ?  12.  Measure  the  yard  stick  with  the  foot 
rule.  13.  How  many  foot  rules  could  be  made  of  a  yard 
stick,  Belle  ?  14.  How  many  feet  long  is  the  yard  stick  ? 
15.  How  many  feet  are  there  in  one  yard  ?  16.  Which 
is  it  better  to  use  in  measuring  the  length  of  the 
room,  the  foot  rule  or  the  yard  stick  ?  17.  Who  •  can 
name  something  that  is  sold  by  the  yard  ?  18.  The  foot 
rule  is  marked  off  into  little  spaces.  19.  What  are  these 
called  ?  20.  How  many  inches  are  marked  on  the  foot 
rule  ?  21.  How  many  are  there  on  the  yard  stick  ? 
22.  What  are  inches  for?  23.  Why  do  we  want  so 
many  different  measures  to  tell  length  and  distance? 
24.  Did  you  ever  hear  any  one  tell  how  far  it  is  from 
your  home  to  the  school  house  ?  25.  Did  he  tell  it  in 
yards,  inches,  or  feet? 

Use  the  yard  stick  and  the  foot  rule  in  measuring  other 
distances  in  which  there  is  no  remainder.  Measure  the 
distance  between  two  lines  four  feet  apart  with  the  yard 
stick.  Measure  the  part  remaining  with  the  foot  rule. 
Measure  in  the  same  way  five  feet,  seven  feet,  ten  feet, 
eleven  feet,  etc.  Mark  the  feet  on  the  yard  stick.  Meas- 
ure three  feet  six  inches  with  the  yard  stick,  to  show  the 
pupils  how  the  yard  stick  may  be  used  to  find  the  exact 
measure  in  inches,  feet,  and  yards.  Give  practice  in 
measuring  yards,  feet,  and  inches  with  the  foot  rule  and 
the  yard  stick.     Give  practice  in  drawing  lines  one  foot 


PINTS,   QUARTS,   AND  GALLONS  23 

long,  two  feet  long,  and  one  yard  long.  Test  these  by 
measuring.  Give  practice  in  estimating  short  distances. 
Test  these  estimates.  Much  interest  will  be  aroused  by 
measuring  the  height  of  the  children.  Lead  the  pupils  to 
see  the  necessity  of  longer  units  of  measure  to  measure 
greater  distances.  Mention  the  rod  and  the  mile  as  such 
units.  Upon  the  completion  of  this  work,  the  pupils 
should  know  that  there  are  twelve  inches  in  one  foot,  and 
three  feet  in  one  yard.  They  should  be  able  to  draw  lines 
to  represent  these  measures  with  a  fair  degree  of  accuracy. 
They  should  be  able  to  use  these  measures  in  finding  the 
length  of  the  room,  the  height  of  other  children,  etc. 

LESSON  VIII  — PINTS,  QUARTS,  AND  GALLONS 

Use  the  pint,  quart,  and  gallon  measures  in  giving  this 
lesson.  The  pupils  should  be  led  to  see  the  necessity  for 
measuring  things  that  are  bought  and  sold.  They  should 
also  see  the  necessity  for  fixed  units  of  measure,  and  for 
units  of  measure  of  various  sizes.  They  should  see  that 
such  indefinite  units  as  half  a  basketful,  half  a  sackful, 
half  a  pailful,  etc.,  cannot  be  fixed  units  for  buying  and 
selling. 

Before  taking  up  this  lesson  the  teacher  should  provide 
a  definite  quantity  of  water,  say  eleven  pints.  Illustra- 
tion. Problems :  1.  Harry,  how  much  water  is  there  in 
this  pail  ?  2.  Alice,  how  much  water  do  you  think  there 
is  in  this  pail  ?  3.  How  much  do  you  think  there  is, 
John?  4.  How  can  we  find  out  how  much  water  there 
is  in  it  ?  5.  What  shall  we  measure  it  with  ?  6.  Select 
one  of  the  measures  for  this  purpose.  7.  This  is  called 
the  pint  measure.  8.  Is  there  a  pint  of  water  in  the  pail  ? 
9.    Are  there  two  pints  of  water  in  it  ?     10.    How  many 


24  PRELIMINARY   LESSONS 

pints  of  water  do  you  think  there  are  in  it,  George  ? 
11.  Measure  it.  Mary  may  write  the  number  of  pints 
on  the  board.  12.  Could  we  measure  it  in  anything  but 
pints  ?  13.  Select  the  quart  measure.  14.  Do  you  think 
there  is  enough  water  to  fill  this  measure  ?  15.  How 
many  quarts  of  water  do  you  think  there  are  in  the  pail  ? 
16.  Measure  it  in  quarts.  17.  Do  you  think  there  is 
enough  water  to  fill  this  larger  measure  ?  18.  Will  it  fill 
it  twice  ?  19.  Who  can  tell  what  this  large  measure  is 
called  ?  20.  Measure  it  with  the  gallon  measure.  21.  Is 
there  enough  left  over  to  fill  the  pint  measure  ?  22.  Ls 
there  enough  left  over  to  fill  the  quart  measure  ?  23.  Fill 
the  quart  measure.  24.  Is  there  a  pint  of  water  left  in 
the  pail  ?  25.  Fill  the  pint  measure.  There  was  one 
gallon,  one  quart,  and  one  pint  of  water  in  the  pail. 
26.  What  is  sold  by  the  pint  ?  By  the  quart  ?  By  the 
gallon  ? 

The  pupils  should  become  familiar  with  the  names 
of  the  measures  through  hearing  them  used.  They 
should  learn  the  relative  sizes  of  the  measures  by  actual 
experience  with  them,  by  using  one  as  a  measure  in  filling 
the  other,  etc.  The  ratio  of  the  several  measures  should 
also  be  perceived.  A  quart  of  water  is  twice  as  much 
as  a  pint  of  water.  Its  ratio  to  a  pint  is  two.  A  pint  of 
water  is  one  half  as  much  as  a  quart  of  water.  Its  ratio 
to  a  quart  of  water  is  one  half.  If  this  •  stands  for  a 
pint  of  water,  what  will  stand  for  a  quart  of  water  ? 
If  J  stands  for  a  quart  of  water,  what  will  stand  for  a 
gallon  of  water?  Use  counters  to  represent  the  measures, 
so  the  pupils  can  handle  them. 

At  the  completion  of  the  work  suggested  in  this  lesson 
the  pupils  should  be  familiar  with  the  pint,  the  quart,  and 
the  gallon  as  units  of  liquid  measure.     They  should  know 


FORMS 


26 


that  a  quart  equals  two  pints,  that  a  gallon  equals  four 
quarts,  that  a  gallon  equals  eight  pints,  that  a  pint  equals 
one  half  of  a  quart,  that  a  quart  equals  one  fourth  of  a 
gallon. 

LESSON   IX  — FORMS 

Teach  pupils  to  distinguish  the  following  geometrical 
forms  :  the  circle,  the  square,  the  oblong,  and  the  triangle. 
To  guard  against  the  common  error  of  associating  these 
names  with  the  lines  which  form  the  perimeters  of  these 
surfaces  instead  of  with  the  surfaces  themselves,  attention 
should  be  directed  to  surfaces  which  stand  out  in  contrast 
to  surrounding  areas.  If  the  blackboard  is  used  to  show 
these  forms,  the  surfaces  sliould  be  shaded,  thus  : 


Cut  circles,  squares,  oblongs,  and  triangles  out  of  card- 
board, and  have  the  pupils  tell  what  the  forms  are.  Have 
the  pupils  draw  these  forms  on  paper.  These  surfaces 
should  be  shaded. 

Problems  :  1.  What  is  the  shape  of  the  window  panes  ? 
2.  What  is  the  shape  of  the  door  ?  3.  What  is  the  shape 
of  the  blackboard  ?  4.  What  is  the  shape  of  the  top  of 
your  desk  ?  5.  What  is  the  shape  of  this  piece  of  paper  ? 
6.  What  is  the  shape  of  this  book  ?  7.  What  is  the 
shape  of  the  top  of  this  table  ?  8.  What  is  the  shape 
of  this  picture  ?  9.  Name  something  in  the  room  that 
is  a  square.  10.  Name  something  that  is  a  triangle. 
11.  Name  something  that  is  a  circle.     12.  A  square  has 


26  PRELIMINARY  LESSONS 

how  many  sides?     13.  A  triangle  has  how  many  sides? 

14.  The  four  sides  of  a  square  are  of  the  same  length. 

15.  Two  sides  of  the  oblong  are  longer  than  the  other  two 
sides.  16.  A  triangle  has  how  many  corners  ?  17.  The 
square  and  the  oblong  have  each  how  many  corners  ? 
18.  Cut  a  square  out  of  paper.  19.  Fold  it  to  make  an 
oblong.     20.  Fold  it  to  make  a  triangle. 

LESSON   X  — FRACTIONS 

One  Half.  1.  To  cut  an  apple  in  half,  we  cut  it 
"through  the  middle."  2.  There  are  two  halves  in  one 
apple.  3.  The  halves  are  of  the  same  size.  4.  If  I  eat 
one  half  of  an  apple,  how  much  of  the  apple  will  be  left  ? 
5.  If  I  cut  two  apples  in  half,  how  many  halves  shall  I 
have  ?  6.  Draw  an  apple  on  the  board.  Using  a  crayon 
for  a  knife,  cut  it  into  halves.  7.  Draw  a  pie,  a  loaf,  an 
orange,  etc.,  on  the  board  and  cut  them  in  the  same  manner. 
8.  How  many  halves  are  there  in  an  orange?  A  loaf? 
A  pie  ?  9.  How  many  halves  are  there  in  anything  ? 
10.  Draw  a  line  on  the  board.  Using  a  crayon,  cut  it 
into  halves.  11.  Which  half  is  the  longer?  12.  Show 
where  you  would  cut  the  ruler  in  order  to  cut  it  into  halves. 
13.  Stand  halfway  from  the  desk  to  the  door.  14.  I  have 
two  apples.  How  can  I  give  one  half  of  them  to  Alice  ? 
15.  Draw  two  apples  on  the  board  and  divide  them  thus  : 
(*5  rS.  16.  Show  in  the  same  way  one  half  of  a  group 
of  four  apples,  six  apples,  eight  apples,  ten  apples,  and 
twelve  apples,  using  the  arrangement  of  groups  given  on  the 
flash  cards.  17.  What  is  one  half  of  six  apples  ?  18.  Place 
eight  boys  in  a  line  in  front  of  the  class.  19.  Send  one 
half  of  the  boys  to  their  seats.  20.  Four  boys  are  one  half 
of  how  many  boys  ?     21.    Send  one  half  of  the  remaining 


FRACTIONS 


27 


boys  to  their  seats.  22.  Two  boys  are  one  half  of  how- 
many  boys  ? 

One  Fourth.  1.  Show  one  half  of  a  circle.  2.  Show 
one  half  of  one  of  these  halves.  3.  Show  one  half  of  the 
other  half  of  the  circle.  4.  The  circle  is  now  cut  into 
four  equal  parts.  5.  Each  part  is  one  fourth  of  a  circle. 
6.  One  fourth  of  a  circle  is  the  same  as  a  quarter  of  'a 
circle.  7.  There  are  four  quarters  or  fourths  in  a  circle. 
8.  There  are  two  fourths  in  one  half  of  a  circle.  9.  In 
the  same  way  show  the  fourths  of  a  square ;  of  an  oblong ; 
of  a  line.  10.  If  I  cut  a  pie  into  fourths,  and  eat  one  of 
the  fourths,  how  many  fourths  will  be  left?  11.  Take 
twelve  blocks  and  divide  th6m  into  four  equal  groups. 
12.    How  many  blocks  are  one  fourth  of  twelve  blocks  ? 

One  Third.  Treat  in  a  similar  manner  the  fraction 
one  third.  Use  the  foot  rule  and  the  yard  stick  in  illus- 
trating one  third.  Show  one  third  of  lines,  circles,  squares, 
and  oblongs.  By  means  of  diagrams  lead  the  pupils  to 
see  that  one  third  of  a  circle  is  less  than  one  half  of  a  circle. 
Using  six  objects,  show  what  is  meant  by  one  half  of  six  ; 
and  with  six  similar  objects  show  what  is  meant  by  one 
third  of  six.    Show  also  what  is  meant  by  two  thirds  of  six. 

Draw  figures  to  illustrate  one  fourth,  three  fourths, 
one  third,  two  thirds,  etc.,  using  squares,  circles,  lines, 
and  oblongs,  and  have  the  pupils  recognize  the  fraction 
by  the  shaded  areas.  With  objects  have  pupils  find 
one  half,  one  fourth,  and  one  third  of  numbers  through 
twelve. 

Using  the  groups  on  the  flash  cards,  have  the  pupils 
show  one  half  of  ten  oranges;  one  half  of  four  apples, 
thus : 


•  •    •  • 


28  PRELIMINARY  LESSONS 

Require  exact  statements  with  reference  to  these 
fractional  parts. 

At  the  completion  of  this  work  the  pupils  should  have 
a  very  clear  conception  of  wliat  is  meant  by  one  half,  one 
third,  and  one  fourth.  They  should  be  a])le  to  show  this 
by  means  of  sim[)le  di-awings.  They  sliould  also  be  able 
t6  show  what  is  meant  by  one  half  of  six  apples,  etc. 
They  should  have  memorized  some  of  the  simple  facts, 
such  as :  two  apples  are  one  half  of  four  apples  ;  three 
boys  are  one  half  of  six  boys  ;  one  half  of  eight  blocks  is 
four  blocks ;  etc. 


LESSON   XI  — DIFFERENCE 

The  purpose  of  this  lesson  is  to  lead  the  pupils  to 
measure  tlie  excess  of  one  quantity  over  another  quantity, 
and  to  acquaint  them  with  the  language  forms  used  to 
express  this  difference.  A  new  operation  is  involved  in 
this,  namely,  subtraction.  Illustration.  Problems: 
1.  Harry,  Walter,  and  Ethel  may  stand  in  line  in  front 
of  the  class.  2.  How  many  children  are  standing  in  line, 
Edna  ?  3,  How  many  girls  are  standing  in  line,  Fred  ? 
4.  How  many  boys  are  standing  in  line,  Mary?  5.  Of 
whicli  are  there  more,  girls  or  boys  ?  6.  How  can  I  make 
the  number  of  boys  and  girls  the  same  ?  7.  Who  can  tell 
me  another  way  of  making  the  number  of  girls  and  boys 
the  same  ?  (The  teacher  should  lead  the  pupils  to  see  that 
the  number  of  boys  and  girls  may  be  made  tlie  same,  (a)  by 
sending  one  of  the  boys  to  his  seat,  or  (h)  by  having  an- 
other girl  stand  in  line.)  8.  There  are  now  two  boys  and 
two  girls  standing  in  line.  If  I  place  another  girl  in  the 
line,  how  many  girls  will  there  be  then  ?  9.  Rose  niay 
stand  in  line  with  the  others.     There  are  now  two  boys  and 


NUMBER  STORIES  29 

three  girls  standing  in  line.  10.  How  many  more  girls  are 
thtire  than  boys  ?  Ayiswer :  There  is  one  more  girl  than 
boys.  11.  Show  this  by  taking  away  one  girl.  12.  Return 
the  girl  to  her  place.     Show  the  same  by  adding  one  boy. 

Measure  differences  between  the  number  of  blocks  in 
two  groups. 

Draw  on  the  board  two  circles  to  represent  two  plates. 
One  of  them  is  the  teacher's,  and  the  other  the  pupil's. 
Place  three  oranges  on  one  plate,  and  one  on  the  other 
plate.  Lead  the  pupils  to  see  how  many  more  oranges 
there  are  on  one  plate  than  on  the  other.  Tlie  number 
that  must  be  added  to  the  smaller  quantity  or  taken  from 
the  larger  quantity,  in  order  to  make  both  quantities  the 
same,  is  the  measure  of  difference.  The  pupils  should  be 
able  to  get  this  difference  by  both  methods.  Make  no 
attempt  to  have  the  pupils  memorize  these  results. 

LESSON   XII  — NUMBER   ST0RIH:S 

Illustration.  Problems:  1.  Harry,  take  five  blocks 
out  of  the  box  and  put  them  on  the  table.  2.  Mary,  tell 
what  Harry  did.  Story:  Harry  took  five  blocks  out  of 
the  box  and  placed  tiiem  on  the  table.  3.  Grace,  you 
may  take  away  two  of  the  blocks  that  are  on  the  table. 

4.  Fred  may  tell  us  a  story  about  what  he  saw.  Story : 
There  were  five  blocks  on  the  table.  Grace  took  two  of 
the  blocks  away.     There  are  three  blocks  left  on  the  table. 

5.  Walter  may  take  two  of  the  blocks  away.  6.  Martha, 
tell  a  story  about  what  you  saw.  Story :  There  were 
three  blocks  on  the  table.  Walter  took  two  of  the  blocks 
away.  There  is  one  block  left  on  the  table.  7.  George 
may  take  one  block  away.  8.  Lottie,  tell  a  story  about 
what  you  saw.     Story :  Tliere  was  one  block  on  the  table. 


30  PRELIMINARY   LESSONS 

George  took  it  away.  There  are  no  blocks  left  on  the 
table.  9.  Alice  may  place  three  blocks  on  the  table. 
10.  Fred  may  tell  what  he  saw.  Story:  There  were  no 
blocks  on  the  table.  Alice  placed  three  blocks  on  the 
table.  There  are  three  blocks  on  the  table.  11.  Jane 
may  place  two  more  blocks  on  the  table.  12.  Richard 
may  tell  a  story  about  what  he  saw.  Story :  There  were 
three  blocks  on  the  table.  Jane  placed  two  more  blocks 
on  the  table.     There  are  now  five  blocks  on  the  table. 

13.  Ruth    may    place    one    more    block    on    the    table. 

14.  Jessie  may  tell  a  story  about  what  she  saw.  Story : 
There  were  five  blocks  on  the  table.  Ruth  placed  one 
more  block  on  the  table.  There  are  now  six  blocks  on 
the  table.  15.  We  shall  call  the  blocks  apples,  and  the 
table  a  tree.  Mary  may  tell  us  how  many  apples  are  on 
the  tree.  16.  Willie  may  pick  two  of  the  apples. 
17.  Frank,  tell  a  story  about  what  Willie  did.  Story: 
There  were  six  apples  on  a  tree.  Willie  picked  two  of 
them.  There  are  four  apples  left  on  the  tree.  18.  Who 
wants  to  pick  two  more  apples  off  the  tree  ?  Vera  may 
do  so.     19.    Edna  may  tell  a  story  about  what  was  done. 

20.  James  may  pick  one  half  of  the  apples  that  are  left. 

21.  Mary  may  tell  what  she  saw.  22.  Fred  may  pick  the 
rest  of  the  apples. 

We  shall  pretend  that  these  two  blocks  are  birds,  and 
the  table  a  fence.  1.  Willie,  tell  a  story  about  what  you 
see.  Story :  I  see  two  birds  sitting  on  a  fence.  2.  Frank, 
show  two  more  birds  coming  to  the  fence.  3.  Grace,  tell 
the  story  about  these  birds.  Story :  There  were  two  birds 
sitting  on  a  fence.  Two  more  birds  came  to  the  fence. 
There  are  four  birds  on  the  fence.  4.  Lucy  may  make  two 
of  the  birds  fly  away.  5.  David  may  tell  a  story  about  the 
birds.     Story:   There  were  four  birds  on  a  fence.     Two 


NUMBER   STORIES  31 

birds  flew  away.  There  are  two  birds  left  on  the  fence. 
6.  Edna  may  bring  three  more  birds  to  the  fence.  7.  Walter 
may  tell  the  story.  Story:  There  were  two  birds  on  a 
fence.  Three  more  birds  flew  to  the  fence.  There  are 
five  birds  on  the  fence.     8.  We  can  tell  this  story  on  the 

2  birds 
board  in  this  way  :    +  3  birds.     The  mark  +  means  more. 

5  birds 
It  means  that  three  more  birds  came  to  the  fence.  9. 
Looking  at  the  board,  Fred  may  tell  us  the  story  again. 
10.  Now  there  are  five  birds  on  the  fence.  Willie  may  help 
two  of  the  birds  to  fly  away.  11.  Grace  may  tell  the 
story  about  the  birds.     12.   We  can  tell  this  story  on  the 

5  birds 
board  in  this  way :   —  2  birds.     13.  Looking  at  the  board, 

3  birds 
Mary  may  tell  the  story  again.     14.  The  mark  —  means 
less.     It  means  that  there  were  two  less  birds  on  the  fence 
after  the  two  birds  flew  away.     15.  I  am  going  to  write  a 
different  story  on  the  board,  and  I  want  to  see  who  can 

2  birds 
tell  it.     +  4  birds.     Story :  There  were  two  birds  on  a 

6  birds 
fence.     Four  more  birds  came  to  the  fence.     There  were 
then  six  birds  on  the  fence.     16.   Who  can  give  the  story 

4  birds 
that  this  tells  ?    —  3  birds.    Story :  There  were  four  birds 

1  bird 
on  a  fence.     Three  of  the  birds  flew  away.     There  was 
one  bird  left  on  the  fence.     17.  Who  can  give  the  story 

2  boys 
that  this  tells  ?     +  2  boys.     This  lesson,  like  all  other 
4  boys 


32  PRELIMINARY   LESSONS 

lessons  in  this  chapter,  should  be  expanded  by  the  teacher. 
Objects  should  be  used  to  illustrate  all  of  the  problems 
given  under  this  lesson,  and  the  groups  should  not  exceed 
six  objects.  The  operation  involved  in  each  i)roblem  is 
thus  made  evident  to  the  pupils.  They  should  be  made 
familiar  with  the  language  forms  associated  with  these 
operations.  They  should  also  learn  to  express  the  opera- 
tions in  such  written  forms  as  are  given  in  Problems  8  and 
12.  Given  the  written  forms,  they  should  be  able  to  make 
number  stories  (problems)  from  them.  They  should  be 
able  to  illustrate  these  by  means  of  objects.  No  attempt 
should  be  made  to  have  the  pupils  memorize  the  results  in 
any  of  the  problems. 

LESSON   XIII —  MEASURE   OF  TIME 

60  seconds  (sec.)  =  1  minute  (min.) 
60  minutes  =  1  hour  (hr.) 

24  hours  =  1  day  (da.) 

7  days  =  1  week  (wk.) 

12  months  (mo.)  =  1  year. 

Teach  the  names  of  the  days  of  the  week.  Use  these 
names  in  written  work. 

Teach  the  names  of  the  months,  their  abbreviations, 
and  the  number  of  days  in  each.  The  number  of  days  in 
each  month  should  be  taught  without  the  use  of  rhyme. 

The  pupils  should  be  able  to  tell  whicli  is  the  sixth 
month,  the  third  month,  what  month  July  is,  October  is, 
etc.,  without  naming  the  months  from  the  beginning  of 
the  year. 

Pupils  should  be  able  to  tell  the  time  of  day,  and  to 
estimate  without  the  use  of  a  timepiece  the  length  of 
seconds  and  minutes. 


SUMMARY  33 

1  .  . 

SUMMARY   OF   CHAPTER   I 

The  work  outlined  in  Chapter  I  should  be  completed 
before  the  book  is  placed  in  the  hands  of  the  pupils.  Upon 
the  completion  of  this  work  the  pupils  should  be  able ; 

1.  To  count  serially  in  any  part  of  the  number  scale 
below  1,000 ;  to  tell  from  the  study  of  the  number  scale 
what  one  more  than  any  number  is,  and  what  one  less  than 
any  number  is  ;  to  tell  the  number  of  tens  there  are  in 
20,  30,  etc. ;  to  tell  what  the  sum  is  when  any  number  less 
than  ten  is  added  to  10,  20,  etc. ;  to  count  by  tens  to  110, 
beginning  with  any  of  the  first  ten  numbers  in  the  scale ; 
to  count  by  fives  to  110,  beginning  with  5,  and  by  twos  to 
50,  beginning  with  2. 

2.  To  read  and  write-numbers  of  two  periods  readily  ;  to 
tell  the  place  value  of  figures  in  numbers  less  than  1,000. 

3.  To  measure  short  distances  in  inches,  feet,  and  yards ; 
to  draw  lines  to  represent  these  units  of  measure  with  a 
fair  degree  of  accuracy ;  to  estimate  lengths  of  less  than 
two  feet  in  inches,  of  less  than  ten  feet  in  feet,  and  of  less 
than  five  yards  in  yards  ;  to  compare  the  length  of  the 
inch,  foot,  and  yard ;  to  use  the  pint,  the  quart,  and  the 
gallon  measures  in  measuring  liquids  ;  to  give  the  number 
of  pints  there  are  in  a  quart,  and  the  number  of  quarts 
there  are  in  a  gallon. 

4.  To  recognize  the  circle,  the  square,  and  the  oblong. 

5.  To  show  what  is  meant  by  one  half,  one  third,  and 
one  fourth ;  to  find  these  fractional  parts  of  small  groups 
of  objects. 

6.  To  measure  groups  of  objects  using  given  units  of 
measure. 

7.  To  solve  simple  problems  objectively,  and  to  express 
the  solution  correctly  in  both  the  oral  and  the  written  form. 

IsT  Bk  AniTii— .'i 


CHAPTER  II 
GENERAL  INTRODUCTION 

ADDITION 

Step  A.  Five  addition  combinations  are  made  to  con- 
stitute a  lesson  in  addition.  These  should  be  perfectly 
memorized.  Sufficient  drill  should  be  given  to  make  them 
perfect  reflexes.  Following  each  new  set  of  combinations, 
on  the  same  page  are  a  number  of  oral  problems,  in  solv- 
ing which  the  pupils  may  refer,  if  necessar}^,  to  the  com- 
binations at  the  top  of  the  page.  Many  similar  problems, 
involving  the  same  combinations,  should  be  given  by  the 
teacher,  and  the  pupils  should  be  encouraged  to  make 
simple  problems  involving  the  combinations  of  each  les- 
son. These  should  deal  with  things  within  the  experience 
of  the  pupils. 

After  the  pupils  have  memorized  the  combinations  of 
a  lesson,  as  given  with  answers  in  the  text,  they  should 
take  up  the  study  of  the  lesson  as  arranged  without  an- 
swers on  the  succeeding  page.  This  study  should  be 
pursued  as  indicated  on  pp.  48,  49.  The  combinations 
should  be  written  on  the  board,  and  the  pupils  should  be 
given  a  rapid  and  thorough  drill  upon  them.  The  order  in 
which  they  are  arranged  on  the  board  should  vary,  and 
the  pupils  should  not  depend  for  their  knowledge  of  the 
combinations  upon  any  sequence  in  the  drill  exercises. 

The  pointer  should  always  rest  upon  the  upper  number 
of  the  combination,  as  the  pupils  will  commence  at  the 

84 


ADDITION  35 

foot  of  the  columns  to  add.  The  pupils  should  give  the 
sums  without  naming  the  numbers  in  the  combinations. 
In  speaking  of  a  combination,  the  lower  number  should  be 
named  first.  The  combinations  of  Lesson  A  are  2  and  3, 
5  and  4,  etc.  The  lesson  should  be  thoroughly  mastered 
through  Step  A  before  the  pupils  undertake  to  study  it  as 
directed  in  Step  B. 

Step  B.  After  the  combinations  of  a  lesson  have  been 
memorized  and  have  been  used  in  simple  problems,  a  study 
exercise  is  provided.  The  teacher  should  see  that  these 
exercises  are  diligently  and  correctly  studied  as  explained 
on  pp.  48,  49.  The  teacher  should  study  the  lessons 
with  the  pupils  until  they  are  able  to  continue  without 
assistance. 

The  purpose  of  this  study  exercise  is  to  prepare  the 
pupils  for  cjplumn  addition.  Nowhere  in  column  addition, 
except  in  the  first  combination  added,  will  the  combina- 
tions occur  as  they  were  memorized  in  Step  A.  In  the 
addition  of  the  column,  the  combination  is  no  longer  2  and 
3,  but  is,  instead,  12  and  3,  22  and  3,  etc.  At  each  step 
in  the  addition  of  a  column,  after  the  first  sum  has  been 
obtained,  the  lower  number  is  retained  mentally.  For 
tliis  reason,  in  studying  the  lessons  through  Step  B,  the 
pupil  is  required  to  retain  mentally  the  tens  of  the  lower 
number  in  each  combination.  Oral  and  written  exercises 
in  which  the  lower  number  is  increased  by  10,  20,  30, 
etc.,  to  100,  may  supplement  this  study,  but  should  not 
supplant  it.-^ 

It  should  be  noted  that  the  combinations  as  they  occur 
in  column  addition  are  in  the  form  24  and  5,  and  not  5 
and  24.  For  this  reason  the  former  should  be  the  form 
used  by  the  teacher  in  both  oral  and  written  combinations. 
After  studying  a  lesson  through  Step  B,  the  pupils  should 


36  GEJSIERAL  INTRODUCTION 

be  able  to  answer  readily,  regarding  each  combination,  such 
questions  as  the  following  for  the  combination  5  and  4  : 
How  many  are  5  and  4?  How  many  are  35  and  4?  65  and 
4  ?  95  and  4  ?  If  a  4  is  added  to  a  number  ending  in  5, 
the  answer  will  end  in  what  number  ?  Before  taking  up 
the  addition  columns,  it  is  expected  that  the  teacher  will 
give  oral  drills  of  this  kind  upon  all  of  the  combinations  of 
a  lesson  after  they  have  been  studied  through  Step  B. 

Step  C.  As  the  combinations  are  arranged  in  the 
several  lessons,  the  sum  of  the  first  combination  is  made 
the  lower  number  in  the  second  combination,  the  sum  of 
the  second  combination  is  made  the  lower  number  in  tlie 
third  combination,  etc.  This  arrangement  is  for  the  pur- 
pose of  constructing  addition  columns  involving  the  com- 
binations of  the  lesson.  With  the  combinations  on  the 
board  for  reference,  arranged  as  in  Lesson  A,  unite  them 
^  n  to  form  short  columns,  thus  : 
ft  f,       Q       o         In  the  combination  2  and  3 

A       A       n       o       I       o         in  the  lesson,  let  the  pointer 

o       o       A        t       ^       ^t         ^^^^  ^^^  ^*     ^^^^^  ^^  ^^  ^^®  ^ 

c       rt       r       r       c^       r         lu  the  first  of  thcsc  columns. 

2       2       5       b       2       b  . 

—      —      —      —      —      —         Ihe  pupil  will  recognize  the 

combination  and  will  give  the  sum,  5.     Move  the  pointer 

to  the  4.     If  the  pupil  hesitates,  pass  the  pointer  to  the 

combination  5  and  4  in  the  lesson.      When  the  sum  has 

been  given,  return  to  the  column  and  add.     The  pupil  will 

soon  see  that  the  2  and  3  in  the  column  stand  in  the  same 

relation  to  4  as  the  5  does  in  the  combination  in  the 

lesson.     If  in  the  addition  of  a  column  the  pupil  fails  to 

add  correctly  any  given  combination,  say  17  and  6,  do  not 

ask  him  what  7  and  6  make,  but  drop  the  column  and  take 

up  the  study  of  the  combination  7  and  6  through  Steps  A 

and   B.     Return  to  the  column,  and  if  the  drill  on  the 


SUBTRACTION  37 

single  combination  was  thorough,  the  pupil  will  be  able  to 
add  the  column. 

Step  C  consists  in  adding  the  columns  as  indicated  in 
Step  C,  pp.  48,  49.  The  pupil  should  begin  the  addition 
of  a  column  by  naming  the  sum  of  the  combination  at  the 
foot  of  the  column.  The  columns  should  be  added  with  a 
regular  cadence.  Perfect  knowledge  of  the  combinations, 
together  with  right  habits  of  work,  will  keep  the  pupil 
from  resorting  to  serial  counting  as  a  means  of  finding  the 
sums.  For  the  fixing  of  these  habits,  much  of  the  work 
on  addition  for  the  first  few  lessons  should  be  oral. 


SUBTRACTION 

Each  lesson  in  addition  is  followed  by  a  corresponding 
lesson  in  subtraction,  and  the  same  combinations  are 
involved  in  both.  Subtraction  is  the  operation  of  finding 
the  difference  between  two  quantities.  This  difference 
may  be  found  by  taking  from  the  greater  of  the  two 
quantities  an  amount  equal  to  the  lesser  quantity  ;  or  it 
may  be  found  by  adding  to  the  lesser  quantity  such  an 
amount  as  will  make  it  equal  to  the  greater.  The  amount 
remaining  in  the  one  case  is  the  amount  added  in  the  other 
case.  Applied  to  the  solution  of  problems  in  subtraction, 
these  methods  are  as  follows  : 

Problem  1 :  Fred  has  5  marbles.  Walter  has  3  marbles. 
How  many  more  marbles  has  Fred  than  Walter  ? 

Problem  2 :  Fred  had  5  marbles.  He  lost  3  of  them. 
How  many  marbles  has  he  left  ? 

By  the  first  method,  a  quantity  equal  in  amount  to  the 
number  of  marbles  Walter  has  is  taken  from  the  number 
of  marbles  Fred  has,  and  the  remaining  number  is  the 
difference,  or  answer.     This  is  expressed  in  the  language 


38       ^  GENERAL  INTRODUCTION 

of  subtraction:    Three  marbles  from  5  marbles  leave  2 
marbles  ;  or  5  marbles  less  3  marbles  are  2  marbles. 

By  the  second  method,  such  an  amount  is  added  to 
the  smaller  quantity  as  will  make  it  equal  to  the  larger 
quantity.  This  is  expressed  in  some  such  language  as; 
"  3  marbles  and  how  many  marbles  are  5  marbles  ?  "  Or, 
"  3  marbles  and  what  are  5  marbles?"  Or,  "3  marbles  and 
2  marbles  are  5  marbles,"  —  a  form  embodying  the  answer. 
Th§  answer,  2  marbles,  is  part  of  the  number  fact  involved 
in  the  addition  form  previously  learned,  when  3  marbles  and 
2  marbles  were  added  to  make  5  marbles.  In  the  second 
problem,  in  which  the  difference  between  a  given  quantity 
and  a  part  of  itself  is  required,  there  is  no  essential  differ- 
ence in  the  way  of  finding  the  part  remaining. 

The  method  of  subtraction  recommended  for  use  in  con- 
nection with  the  exercises  of  this  text  is  the  second  of  the 
above  methods,  which  is  the  so-called  "Austrian  method." 
It  ii  also  known  as  the  "additive  method,"  and  is  some- 
times spoken  of  as  the  "  computors'  method "  or  the 
"method  of  making  change."  Its  advantage  lies  in  the 
fact  that  the  number  facts  of  addition  are  used  to  find 
the  differences  in  subtraction.  After  the  addition  com- 
binations have  been  memorized,  no  new  number  facts  are 
necessary  in  order  to  perform  the  operation  of  subtraction. 
The  pupil  has  only  to  learn  how  to  apply  to  a  new  mode 
of  expression  the  knowledge  that  he  has  already  acquired. 
This  he  learns  to  do  without  much  difficulty. 

3 

The  addition  combination  2  is  read,  2  and  3  are  5.     The 

6 
5 
corresponding  subtraction  —  2  is  read,  2  and  what  are  6, 

3 
or  2  and  how  many  are  5,  or  simply  2  and  3  are  6. 


SUBTRACTION  39 

Subtract  thus:  5  and  4  are  9;  3  and  2  are  5.  59 
Write  the  4  under  the  5  and  the  2  under  the  3.  —35 
That  is,  supply  the  figure  required  in  each  column  24 
in  order  to  obtain  the  sum  at  the  top. 

If  the  figure  in  the  subtrahend  represents  an  amount 
larger  than  the  corresponding  figure  in  the  minuend,  sub- 

52 
tract  thus :   —  29,  9  and  3  are  12 ;  carry  1  to  2  as  in  addi- 

23 
tion,  making  it  3 ;  3  and  2  are  5.     The  answer  is  23. 

This  method  of  subtraction  may  also  be  explained  as 
follows:  5        15        25 

Compare  —3,  —13,  —23.  If  both  the  minuend  and 
~2  "2  "2 
the  subtrahend  are  increased  by  the  same  amount,  the 
difference  is  not  changed.  The  steps  necessary  to  find  the 
difference  may  be  explained  thus :  It  is  evident  that  there 
is  no  number  (excepting  a  negative  quantity)  which  added 
to  9  makes  2  ;  so  10  is  added  to  the  2,  changing  that 
number  to  12.  Nine  and  3  make  12.  Ten  must,  therefore, 
be  added  to  the  subtrahend  to  equalize  this  change.  This 
is  done  by  increasing  the  next  lower  number  by  1.  This 
changes  the  2  to  3.    Three  and  2  make  5.    The  answer  is  23. 

This  may  be  given  :  9  from  12  leaves  3.  3  from  5 
leaves  2,  if  the  teacher  prefers  to  use  this  language  form 
to  express  the  operation.  The  two  language  forms  should 
not  be  confused.  The  2  in  the  minuend  is  increased  by 
the  addition  of  10,  and  the  lower  number  in  the  next 
combination  is  increased  by  1  to  equalize  this  change,  as 
above. 

Again,  the  minuend  is  the  sum  of  two  numbers.  The 
subtrahend  is  one  of  the  numbers.  The  other  number  is 
a  number  which  added  to  the  subtrahend  will  make  the 
minuend. 


40  GENERAL  INTRODUCTION 

(  )  difference,  or  other  addend.  72  minnend. 

27  subtrahend,  or  given  addend.       —  27  subtrahend. 
72  minuend,  or  sum  of  addends.  45  difference. 

Explanation:  7  and  5  make  12.  Write  5  as  the  units' 
figure  in  the  missing  addend,  and  add  the  1  ten  to  the 
2  tens  of  the  other  addend.  3  and  4  make  7.  45  is  the 
missing  addend. 

The  method  of  subtraction  involving  the  "  borrowing  " 
of  1  ten  from  the  next  place  in  the  minuend,  etc.,  should 
not  be  used  in  the  exercises  of  this  text,  as  the  combina- 
tions would  thereby  be  changed.  Do  not  permit  pupils  to 
acquire  the  habit  of  making  these  changes  in  the  written 
work. 

MULTIPLICATION 

Facility  in  multiplication  and  division,  as  in  addition 
and  subtraction,  is  acquired  only  by  much  practice.  Each 
lesson  in  multiplication  consists  of  a  few  number  facts 
which  are  to  be  memorized  perfectly,  and  then  used  until 
accuracy  and  facility  in  handling  these  have  been  acquired. 
The  number  facts  of  each  lesson  should  become  perfect 
reflexes  before  the  new  facts  of  the  succeeding  lesson  are 
introduced.  Much  drill  work  is  provided  for  in  the  text. 
This  should  be  supplemented  by  simihir  material  in  case  the 
pupils  have  not  acquired  the  desired  skill  in  handling  the 
facts  of  each  lesson.  No  provision  has  been  made  for 
the  mastery  of  the  "  tables  "  as  such.  Each  fact  must  finally 
be  known  without  reference  to  the  table  to  which  it  belongs. 
A  table  of  products  and  quotients  is  given  near  the  close 
of  the  chapter  on  multiplication  and  division.  It  is  for 
reference  merely.  Such  multipliers  as  20,  30,  etc.,  are 
introduced  before  multipliers  like  23,  34,  etc.,  since  the 


DIVISION  41 

arrangement  of  the  product  with  the  first  class  of  multi- 
pliers is  more  nearly  like  that  with  multipliers  of  one  place. 
Facility  in  the  mechanical  work  of  multiplication  re- 
quires not  only  a  ready  knowledge  of  the  several  products, 
but  also  the  ability  to  find  readily  the  sum  of  the  product 
and  some  given  number, — tlie  number  to  be  "carried." 
This  is  provided  for  in  the  drill  exercises  that  follow  each 
set  of  facts  in  multiplication.  This  furnishes  a  constant 
review  of  the  addition  combinations  as  well  as  the  desired 
practice  in  adding  a  number  to  each  product. 

DIVISION 

Each  lesson  in  multiplication  is  followed  by  a  corre- 
sponding lesson  in  division.  The  study  exercises  in 
both  should  be  thoroughly  mastered  before  attempting 
the  exercises  which  follow.  With  one  place  divisors, 
short  division  is  used.  The  pupils  should  acquire  the 
habit  of  using  the  shorter  method  with  divisors  of  one 
place.  The  several  steps  in  long  division  are  taught 
when  occasion  arises  for  their  use.  A  special  method  is 
employed  in  the  subject  of  long  division.  Much  time  is 
usually  wasted  by  the  pupil  in  the  attempt  to  find  the 
correct  quotient  figure,  with  the  result  that  pupils  become 
discouraged  and  acquire  a  dislike  for  the  work.  A  method 
is  here  presented  by  which  the  correct  quotient  figure  may 
be  found  with  little  difficulty.  (See  pp.  178-186.)  The 
teacher  should  acquaint  herself  with  the  method  and  should 
see  that  the  pupils  are  perfectly  familiar  with  it  before 
they  are  required  to  use  it.  Time  spent  in  mastering  it 
will  be  time  saved.  A  study  of  divisors  with  reference 
to  determining  the  quotient  figure  is  given  in  the  text. 
The  pupils  should  be  taught  to  inspect  closely  each  new 


42  GENERAL  INTRODUCTION 

divisor  before  attempting  to  use  it.  Use  only  such  divisors 
as  properly  belong  in  each  of  the  classes  into  which  they 
are  grouped  in  the  text. 

A  divisor  of  one  place  is  used  in  presenting  the  several 
steps  in  long  division.  This  is  accompanied  by  a  second 
illustration  in  which  the  divisor  consists  of  two  places. 
The  authors  are  quite  certain  that  many  teachers  will 
secure  better  results  by  teaching  the  steps  in  long  divi- 
sion in  connection  with  divisors  of  two  places.  Until  the 
process  has  been  thoroughly  learned,  use  as  divisors  num- 
bers in  which  the  second  figure  is  the  same  as  or  less  than 
the  first  figure  ;  as,  44,  55,  84,  32,  43,  75,  97,  etc. 


CHAPTER   III 
ADDITION  AND   SUBTRACTION 

NOTATION,  NUMERATION,  OBJECTIVE  FRACTIONS,  COM- 
POUND  NUMBERS,  MULTIPLICATION,   AND  DIVISION 

I.  1.    One  ten  and  two  ones  are  how  many  ones  ? 

2.  One  ten  and  seven  ones  are  how  many  ones  ? 

3.  Twelve  ones  are  one  ten  and  —  ones. 

4.  Eighteen  ones  are  —  ten  and  —  ones. 

5.  Twenty  ones  are  —  tens. 

6.  Two  tens  and  three  ones  are  —  ones. 

7.  Twenty-five  ones  are  —  tens  and  —  ones. 

8.  Two  tens  and  seven  ones  are  —  ones. 

9.  Three  tens  are  —  ones.     Four  tens  are  —  ones. 
10.  Eighty  ones  are  how  many  tens? 

II.  Nine  tens  and  six  ones  are  —  ones. 

12.  Seventy-eight   ones  are  —  tens  and  —  ones. 

13.  Ten  ones  are  —  ten.     Ten  tens  are  —  ones. 

14.  How  many  tens  are  there  in  seventy  ones  ? 

15.  How  many  tens  are  there  in  eighty  ones  ? 

16.  Sixty  ones  are  —  tens.     Eighty  ones  are  — 
tens. 

43 


44  ADDITION   AND   SUB  IRACTION 

2.  1.    Five  tens  and  no  ones  are  —  ones. 

2.  Five  tens  and  nine  ones  are  —  ones. 

3.  Twenty-four  ones  are  twenty  ones  and  —  ones. 

4.  Thirty-six  ones  are  thirty  ones  and  —  ones. 

5.  Twenty-three  apples  are  twenty  apples  and  — 
apples. 

6.  Thirty-seven  cents  are  thirty  cents  and  —  cents. 

7.  Forty-eight  children  are  forty  children  and  — 
children. 

8.  Sixteen  boys  are  ten  boys  and  —  boys. 

9.  Eighteen  oranges  are  ten  oranges  and — oranges. 

10.  Twenty  girls  and  six  girls  are  —  girls. 

11.  Thirty   marbles    and    eight    marbles    are  — 
marbles. 

12.  Forty  girls  and  ten  girls  are  —  girls. 

13.  Twenty  cents  and  ten  cents  are  —  cents. 

14.  Twenty-four  days  and  ten  days  are  —  days. 

15.  Thirty-six  days  and  ten  days  are  —  days. 

16.  Sixteen  boys  and  ten  boys  are  —  boys. 

17.  Seven  boys  and  ten  boys  are  —  boys. 

18.  Nine  birds  and  ten  birds  are  —  birds. 

19.  Ten  oranges  and  six  oranges  are  —  oranges. 

20.  Ten  books  and  four  books  are  —  books. 

21.  Eight  pencils  and  ten  pencils  are  —  pencils. 

22.  What  number  is  ten  more  than  thirty  ? 

23.  What  number  is  ten  more  than  seventy? 


ADDITION   AND   SUBTRACTION  45 

24.  Thirty-five  is  ten  more  than  — .     Thirty-six  is 
ten  more  than  — . 

25.  Forty-six  is  how  many  more  than  thirty-six  ? 

26.  Sixty-eight  is  how  many  more  than  fifty-eight  ? 

3.  1.    There  are  ~  lOO's  in  400.     There  are  —  I's 
in  400. 

2.  There  are  —  lOO's  in  1000.     Count  by  lOO's 
to  1000. 

3.  What  number  is  one  hundred  more  than  60  ? 

4.  What  number  is  one  hundred  more  than  67  ? 

5.  What  number  is  one  hundred  more  than  167  ? 

6.  What  number  is  one  hundred  more  than  625  ? 

7.  Write  a  number  that  tells  9  tens  and  4  ones. 

8.  Write  a  number  that  tells  2  hundreds,  6  tens, 
and  8  ones. 

4.  1.    What  number  is  one  more  than  6  ? 

2.  What  number  is  one  more  than  16  ? 

3.  What  number  is  one  more  than  13  ? 

4.  What  number  is  one  more  than  20  ? 

5.  What  number  is  one  more  than  29  ? 

6.  What  number  is  one  less  than  4  ? 

7.  What  number  is  one  less  than  9  ? 

8.  What  number  is  one  less  than  20  ? 

9.  What  number  is  one  less  than  29  ? 
10.  What  number  is  one  less  than  40  ? 


46  ADDITION   AND  SUBTRACTION 

5.  Write  in  figures  : 

1.  Twenty  and  eight  are — .     4.  Ten  and  six  are  — . 

2.  Forty  and 'five  are  — .        5.  Fifty  and  two  are  — . 

3.  Eighty  and  nine  are  — .       6.  Forty  and  seven  are — . 

6.  Read  these  numbers  : 

678        390        1,357        7,006         4,800         10,064 
804        862        1,048        8,070         5,630         12,006 

Tell  what  each  figure  in  the  above  numbers  stands 
for. 

7.  Write  in  figures : 

1.  Thirty-eight.  2.  Four  hundred  seven.  3.  Eight 
hundred  sixty.  4.  Nine  hundred.  5.  Three  hundred 
sixty-five.  6.  Seven  hundred  eighteen.  7.  Two  hun- 
dred twelve.  8.  One  thousand  seven.  9.  Four  thou- 
sand three  hundred  twenty,     lo.  Six  thousand  sixty. 

8.  1.  Count  by  tens  to  122,  beginning  with  2. 

2.  Count  by  tens  to  125,  beginning  with  5. 

3.  Count  by  tens  to  129,  beginning  with  9. 

4.  Count  by  tens  to  124,  beginning  with  4. 

5.  Count  by  tens  to  123,  beginning  with  3. 

3  birds 

6.  Give  a  number  story  for  this :  *    +2  birds 

5  birds 
9  boys 

7.  Give  a  number  story  for  this :      —  5  boys 


4  boys 


♦  See  p.  31. 


ADDITION  —  LESSON   A  47 

ADDITION  — LESSON  A 
9.  1.  Memorize  the  folloiving  : 

3  4  3  2  6 

2  5  ^  2  Jt 

5  9  12  4  10 

2.    Give  a  number  story  suggested  by  each. 

Model  :  There  were  2  books  on  the  table.  Mary 
put  3  more  books  on  the  table.  There  were  then 
—  books  on  the  table. 

10.   Oral  Problems.*^ 

1.  Four  boys  and  five  boys  are  —  boys. 

2.  Nine  girls  and  three  girls  are  —  girls. 

3.  Six  apples  and  four  apples  are  —  apples. 

4.  How  many  are  two  books  and  three  books  ? 
Ansioer :  Two  books  and  three  books  are  five  books. 

5.  How  many  are  three  chairs  and  nine  chairs  ? 

6.  How  many  are  four  feet  and  six  feet  ? 

7.  Five  days  and  four  days  are  how  many  days  ? 

8.  Two  hours  and  two  hours  are  how  many  hours  ? 

9.  A  girl  paid  five  cents  for  a  ribbon  and  four 
cents  for  a  tablet.  How  much  did  she  pay  for  both  ? 

The  answer  in  addition  is  called  the  sum. 

*  In  this  and  similar  exercises  the  teacher  is  expected  to  dictate  addi- 
tional problems  involving  the  combinations  of  the  lesson.  Place  the  com- 
binations on  the  board  without  answers  and  have  the  pupils  make  problems 
for  one  another  to  solve.  Correct  answers  should  be  given  in  correct 
language. 


48  ADDITION   AND   SUBTRACTION 

11.   Study  Exercises. 

3  4  3  2  6 

2  5  9  2  4 

9  4  2  2  5 

3  6  2  3  4 


HOW  TO   STUDY  ADDITION* 

After  the  combinations  have  been  memorized,  the  study 
of  the  lesson  in  addition  is  to  be  taken  up  as  indicated  in 
Steps  A,  B,  and  C,  as  follows  : 

Step  A.  Study  the  combinations,  without  answers  as 
above,  until  you  can  give  each  result  without  hesitation. 
In  studying  a  combination,  direct  the  thought  to  the 
upper  number,  as  for  the  present  we  shall  begin  at  the 
foot  of  the  column  to  add.  Pass  from  one  combination  to 
another.  Continue  the  study  until  you  can  give  the  sums 
as  readily  as  you  can  read :  5,  9,  12,  4,  10. 

Step  B.  Mentally  place  1  ten  before  each  of  the  lower 
numbers  in  the  upper  set  of  combinations  and  add:  12, 
15;  15,19;  19,22;  12,14;  14,20. 

Pass  to  the  lower  set  of  combinations,  add  2  tens  to 
the  lower  numbers,  and  add:  23,  32;  26,  30;  22,  24; 
28,  25  ;  24,  29. 

Return  to  the  upper  set,  and  mentally  place  3  tens 
before  each  of  the  lower  numbers,  and  add:  32,  35; 
35,  39;  etc. 

*  The  teacher  should  explain  each  step  carefully.  It  is  important 
that  the  pupils  learn  at  the  start  how  to  study  the  addition  exercises 
correctly. 


HOW   TO   STUDY   ADDITION  49 

Pass  again  to  the  lower  set,  then  return  to  the  upper 
set,  each  time  increasing  the  number  of  tens  until  10  tens 
are  added,  and  the  combinations  add :  103,  112 ;  106, 
110;  etc.  The  columns  to  be  added  are  made  up  of  the 
combinations  studied  in  Step  B. 

Step  C  Begin  at  the  foot  of  the  column  and  add  as  in 
Column  e  below :  5,  9,  12,  14,  20. 

Mentally  place  1  ten  before  the  number  at  the  foot  of 
the  column,  and  add :  12,  15,  19,  22,  24,  30. 

Mentally  place  2  tens  before  the  number  at  the  foot  of 
the  column  and  add.  Increase  the  number  of  tens  until 
the  column  is  added :  102,  105,  etc. 


12.   Oral  Drill.=^ 

Give  the  sums  of : 

1. 

15  and  4.    5.  29  and  3. 

9. 

56  and  4. 

13. 

23  and  9. 

2. 

24  and  5.    6.  22  and  3. 

10. 

74  and  6. 

14. 

42  and  2. 

3. 

19  and  3.    7.  36  and  4. 

11. 

13  and  2. 

15. 

69  and  3. 

4. 

16  and  4.    a  44  and  5. 

12. 

12  and  3. 

16. 

65  and  4. 

13.   Oral  Exercises. 

Add  as  indicated  in  Step  C  above. 


a 

b 

C 

d 

e 

/ 

9 

h 

I 

i 

k 

I 

TO 

n 

6 

5 

3 

4 

2 

2 

2 

3 

2 

3 

4 

6 

2 

6 

6 

3 

3 

3 

3 

3 

3 

5 

6 

4 

3 

3 

2 

2 

3 

4 

5 

4 

9 

5 

2 

4 

6 

4 

4 

4 

3 

3 

9 

3 

2 

5 

6 

2 

2 

6 

4 

3 

3 

3 

4 

4 

6 

3 

9 

4 

2 

3 

6 

4 

6 

2 

2 

2 

5 

5 

4 

9 

3 

6 

2 

9 

4 

6 

4 

*  Oral  drill  should  be  given  upon  all  of  the  combinations  in  each  of  the 
study  exercises  before  taking  up  the  addition  of  the  columns. 

IsT  Hk  a  kit  1 1— 4 


50  ADDITION    AND  SUBTRACTION 


14.  1.   One  whole  is  —  halves. 

2.  One  whole  is  —  fourths. 

3.  One  half  is  —  fourths. 

4.  Two  fourths  and  one  fourth  are  —  fourths. 

5.  One  half  and  one  fourth  are  —  fourths. 

6.  Two  fourths  less  one  fourth  is  —  fourth. 

7.  One  whole  less  one  fourth  is  —  fourths. 

8.  If  a  pie  is  cut  into  two  equal  pieces,  one  of  the 
pieces  is  —  half  of  the  pie. 

9.  If  one  half  of  a  pie  is  cut  into  two  equal  pieces, 
one  of  the  pieces  is of  the  pie. 

10.  A  woman  divided  a  pie  equally  among  some 
children.  There  were  four  children.  Each  child 
received of  a  pie. 

11.  Harry  had  one  half  of  a  pie.  George  had  one 
half  as  much  as  Harry.     George  had of  a  pie. 

12.  Two  wholes  are  —  halves. 

13.  A  woman  had  two  pies.  She  gave  three  boys 
each  one  half  of  a  pie.     How  much  pie  had  she  left  ? 

14.  Fold  a  paper  into  four  equal  folds. 


SUBTRACTION  — LESSON   A  61 

SUBTRACTION  — LESSON  A 

15.  1.  Memorize  the  following ': 

9        12        6        10        4        9        12        5        10 

_4       _9     -2       -4     -2     -5       -3    -3      -6 

5  33  624  924 

The  exercise  at  the  right  is  read,  6  and  how  many 
are  10  ?     The  answer  is  — . 

2.  Read  the  above  exercises,  beginning  at  the  right. 

3.  Give  a  number  story  suggested  by  each. 

Model  :  Fred  had  10  apples.  He  gave  6  of  them 
to  James.     Fred  had  —  apples  left. 

16.  Oral  Problems. 

1.  Four  books  and  —  books  are  nine  books. 

2.  Two  days  and  —  days  are  five  days. 

3.  Nine  months  and  —  months  are  twelve  months. 

4.  Five  chairs  are  three  chairs  and  —  chairs. 

5.  Ten  weeks  are  six  weeks  and  —  weeks. 

6.  Nine  boys  are  four  boys  and  —  boys. 

7.  There  were  five  girls  in  a  room.  Three  of 
them  were  seated.     How  many  were  not  seated  ? 

8.  A  boy  had  five  apples.  He  gave  away  two 
apples.     He  had  —  apples  left. 

9.  There  were  ten  words  on  the  blackboard. 
Lottie  erased  six  of  the  words.  How  many  words 
were  left  on  the  blackboard  ? 


62  ADDITION    AND   SUBTRACTION 

17.  Study  Exercises. 

9        12        5        10        4        9        12        5  10 

_4       ^9     -^2      -4-2-5       1:3-3  -6 

Study  the  above  exercises  until  you  can  give  the 
answers  as  readily  as  you  can  read  these  numbers: 
5,  3,  3,  6,  2,  4,  9,  2,  4. 

18.  Written  Exercises. 

a          b           c          d           e          f          g          h  i 

1.  59   95   55   95   50   90   99   92  44 
-24  -53  -30  -40  -50  -50  -40  -50  -22 

2.  125  109  124  105  129  103  122  100  120 
_93  _44  _32  -60  -30  -43  -92  -40  -90 


19.   Written  Exercises. 

In  Exercise  a  below,  the  9  is  greater  than  the  2 
above  it,  so  we  say  9  and  3  are  12.  We  then  add  1 
to  the  next  lower  number.     This  chan<jjes  it  to  3.     We 


th 

en  say  3  and  2  are  5. 

1 

abed 

52       52      92      92 

-29    -23  -49  -33 

e 
90 

-44 

/ 

50 

-26 

9 

40 
-14 

h 

90 
-36 

i 

50 
-14 

abode  f 

2      $  55    100  in.      90  ft.      50  yd.    102  gal.      44  in. 
-$23  -36  in.  -46  ft.  -30  yd.  -59  gal.  -22  in. 
$  in.  ft.  yd.  gal.  in. 


ADDITION   AND   SUBTRACTION  — LESSON   A         53 

ADDITION   AND   SUBTRACTION  —  LESSON   A 
20.    Oral  Problems. 

1.  A  boy  paid  $8  for  a  pair  of  shoes  and  $2  for 
a  hat.     How  much  did  he  pay  for  both  ? 

2.  Nine  ducks  were  swimming  on  a  pond.  Five 
flew  away.     How  many  ducks  were  left  on  the  pond  ? 

3.  A  boy  threw  at  a  mark.  He  missed  it  6  times 
and  struck  it  4  times.    How  many  times  did  he  throw? 

4.  Twelve  girls  were  invited  to  a  party.  Three 
of  the  girls  were  not  able  to  go.  How  many  girls 
went  to  the  party  ? 

5.  A  boy  spent  part  of  his  vacation  in  camping 
out  and  the  rest  of  it  in  the  city.  He  camped  for 
9  weeks  and  was  in  the  city  3  weeks.  How  long 
was  his  vacation? 

6.  A  girl  had  a  ribbon  5  yards  long.  She  cut  off 
a  piece  2  yards  long.  How  long  was  the  part  that 
was  left  ? 

7.  A  boy  had  10  cents.  He  bought  a  tablet  for 
4  cents.     How  much  money  did  he  have  left  ? 

8.  If  it  takes  2  yards  of  cloth  to  make  an  apron, 
how  many  yards  will  it  take  to  make  two  aprons  ? 

What  is  the  answer  in  addition  called  ? 
The  answer  in  subtraction  is  called  the  difference, 
or  remainder. 


54  ADDITION   AND  SUBTRACTION 

21.    Written  Exercises. 

Begio  with  Exercise  a  and  add  the  right-hand 
column,  thus :  9,  12,  14.  Write  the  4  under  the 
column.  Add  the  1  to  the  1  at  the  foot  of  the 
next  column,  and  add :    2,  4,  9,  12. 


a 

6 

C 

d 

e 

/ 

g 

h 

632 

223 

393 

255 

543 

544 

820 

260 

253 

339 

444 

322 

255 

236 

633 

324 

324 

444 

202 

493 

342 

902 

244 

436 

815 

436 

253 

429 

852 

312 

155 

380 

22.   Written  Exercises. 

a              h             c             d  e  f  g 

1.  925   120   504   422  950  502  900 

-392  -84  -142  -189  -414  -139  -356 


a 

6 

C 

d 

e 

2. 

$520 

$1000 

922 

in. 

502  ft. 

400  yd. 

— 

1286 

-$354 

-419 

in.  - 

-293  ft. 

- 196  yd. 

23 

.  Written  Exercises. 

a 

6 

c 

d 

e 

/ 

g          h 

943 

353 

693 

524 

336 

539 

546   462 

334 

363 

234 

93 

32 

406 

634   323 

635 

924 

363 

633 

493 

642 

236   329 

246 

633 

429 

245 

64 

233 

342   466 

134 

282 

303 

54 

325 

109 

832   324 

ADDITION  AND   SUBTRACTION  — LESSON   A  55 

24.   Written  Problems. 

1.  A  man  bought  a  horse  for  |90  and  a  harness 
for  $30.     How  much  did  he  pay  for  both  ? 

2.  There  were  59  ducks  on  a  pond.  Twenty-five 
of  them  flew  away.  How  many  ducks  were  left  on 
the  pond  ? 

Model  for  Addition  Model  for  Subtraction 

$  90  cost  of  horse  59  ducks  on  pond 

+  $  30  cost  of  harness  -  25  ducks  flew  away 
$120  cost  of  both  34  ducks  left  on  pond 

3.  There  are  23  girls  in  Room  A  and  32  girls  in 
Room  B.     How  many  girls  are  there  in  both  rooms  ? 

4.  There  were  55  pupils  in  a  room.  Thirty-two 
pupils  left  the  room.  How  many  pupils  remained  in 
the  room  ? 

5.  A  boy  had  95  cents.  He  spent  50  cents.  How 
much  money  had  he  left  ? 

6.  A  farmer  had  29  cows.  He  bought  23  cows. 
How  many  cows  did  he  then  have  ? 

7.  Fred  has  23  marbles.  James  has  32  marbles. 
How  many  marbles  have  both  ? 

8.  A  man  bought  a  wagon  for  $  50  and  a  team  of 
horses  for  $  240.     How  much  did  both  cost  him  ? 

9.  A  grocer  sold  62  pounds  of  sugar  on  Monday 
and  43  pounds  on  Tuesday.  How  many  pounds  did 
he  sell  on  both  days  ? 


56 


ADDITION  AND  SUBTRACTION 


One  Pint 


One  Quart 


One  Gallon 


25.  1.    Two  -pints  are  one  quart. ^ 

2.  Four  quarts  are  one  gallon. 

3.  One  pint  is  —  half  of  one  quart. 

4.  One  quart  is  —  times  one  pint. 

5.  Two  pints  and  two  pints  are  —  pints. 

6.  The  ratio  of  a  quart  to  a  pint  is  2. 

7.  The  ratio  of  a  gallon  to  a  quart  is  — . 

8.  One  quart  is  —  fourth  of  a  gallon. 

9.  Two  quarts  are  —  fourths  of  a  gallon. 

10.  Three  quarts  are  —  fourths  of  a  gallon. 

11.  One  pint  and  one  quart  are  —  pints. 

26.  Written  Exercises. 

a  h  c  d 

1200   4529    9240   2295 
-260  -290  -4314   -342 


e 

/ 

3520 

3224 

-186 

-832 

2.  4535 
-2230 


4502    5200 
2493  -2256 


9549   3005   1902 
5240  -543  -340 


♦  Review  Lesson  VIII,  p.  23,  before  taking  up  llie  study  of  tliis  page. 


ADDITION  — LESSON   B  57 

ADDITION  — LESSON  B 

27.    1.   Memorize  the  following  : 

3  4  3  2  2 

6  _9  3  6  _8 

9  13  6  8  10 

2.  Give  a  number  story  suggested  by  each. 

3.  A  boy  had  9  marbles.     He  bought  4  marbles. 
He  then  had  —  marbles. 

4.  Six   oranges   and    3    oranges   are   how   many 
oranges  ? 

5.  Eight  tons  of  coal  and  2  tons  of  coal  are  — 
tons  of  coal. 

6.  There  are  —  3's  in  6.     There  are  —  2's  in  4. 

7.  There  are  —  3's  in  9.     How  do  you  know  this  ? 

8.  Two  hats  and  —  hats  are  10  hats. 

9.  Nine  boys  and  —  boys  are  13  boys. 

10.  Eight  cents  and  —  cents  are  10  cents. 

11.  Harry  has  6  books  and  Willie  has  3  books. 
The  two  boys  together  have  —  books. 

12.  There  are  9  girls  in  the  first  row  and  4  gh^ls  in 
the  second  row.     There  are  —  girls  in  the  two  rows. 

13.  Mary  picked  6  boxes  of  berries  in  the  morning 
and  2  boxes  in  the  afternoon.  She  picked  —  boxes 
during  the  day. 

14.  Ethel  has  two  pieces  of  ribbon.  There  are  4 
yards  in  the  first  piece  and  9  yards  in  the  second. 
In  the  two  pieces  there  are  —  yards. 


58  ADDITION  AND  SUBTRACTION 

28.    Study  Exercises. 

Study  as  indicated  in  Steps  A  and  B,  pp.  48,  49. 

3  4  3  2  2 

6  9  3  6  8 

6  8  3  9  6 

2  2  3  4  3 

To  THE  Teacher.     Give  oral  drill   on  the  com- 
binations in  the  above  study.     See  p.  49. 


29. 

Oral  Exercises. 

Add 

as 

indicated 

in 

Step 

,C, 

p.  49. 

a 

6 

c 

d 

e 

/ 

9 

h 

i 

i 

k 

I 

TO 

71 

2 

8 

3 

2 

3 

4 

4 

2 

3 

4 

6 

6 

8 

3 

2 

2 

4 

3 

4 

9 

3 

3 

4 

9 

4 

2 

2 

3 

3 

2 

3 

4 

9 

2 

3 

4 

9 

2 

6 

2 

6 

4 

4 

3 

3 

3 

2 

2 

3 

9 

2 

2 

4 

3 

2 

6 

3 

4 

9 

3 

2 

3 

2 

2 

6 

3 

6 

9 

8 

9 

6 

9 

4 

3 

6 

3 

8 

8 

2 

3 

3 

4 

2 

4 

DOLLARS   AND  CENTS 

30.    $3.50  is  read  three  dollars  and  fifty  cents. 

$3.60  is  read  three  dollars  and  —  cents. 

$3.05  is  read  —  dollars  and  five  cents. 

$.05  is  read  —  cents.     $.50  is  read  —  cents. 

The  point  (.)  separates  dollars  from  cents. 

Write  the  following  in  a  column,  cents  under  cents 
and  dollars  under  dollars,  and  add : 

$.34,   $2.36,   $.03,  $6.42,  $1.08,  $.30. 


SUBTRACTION  — LESSON  B  69 

SUBTRACTION  — LESSON  B 

31.    1.    Memorize  the  following : 
9       13        8       10         6         9       13        8       10 
_3      -4     -2      -8     -3     -6      -9     -6     -2 
6         96         233         428 

2.  Give  a  number  story  suggested  by  each. 

3.  Two  girls  and  how  many  girls  are  10  girls  ? 

4.  Nine  days  and  how  many  days  are  13  days  ? 

5.  Eight  cents  and  how  many  cents  are  10  cents  ? 

6.  Three  feet  and  —  feet  are  9  feet. 

7.  Four  weeks  and  —  weeks  are  13  weeks. 

8.  Six  yards  and  —  yards  are  8  yards. 

9.  Ten  cents  are  2  cents  and  —  cents. 

10.  Nine  days  are  3  days  and  —  days. 

11.  Thirteen  weeks  are  4  weeks  and  —  weeks. 

12.  A  boy  had  13  weeks'  vacation.  He  spent  the 
first  four  weeks  of  it  in  the  country.  How  many 
weeks  of  his  vacation  were  left  ? 

13.  A  girl  bought  a  2-cent  stamp.  She  handed  the 
postmaster  a  dime.  He  gave  her  back  —  cents 
change. 

14.  There  were  9  girls  in  a  room.  Three  of  them 
were  reading.  The  others  were  drawing.  How  many 
girls  were  drawing  ? 

15.  A  boy  has  13  examples  to  work.  After  he 
has  worked  4  of  them,  how  many  will  he  then  have 
to  work  ? 


60  ADJHTION   AND   SUBTRACTION 


33.   Study  Exercises. 

9       13         8       10 

6 

10 

9 

13 

8 

_3     _4     -6      -2 

-3 

-8 

-6 

-9 

-2 

Study  the  above  exercises  until  you  can  give  the 
answers  without  hesitation. 


33.   Written  Exercises. 

a 
1.     139 
-93   - 

b         c           d           e 

90      98       63       808 

-58   -66   -29   -582 

/        9 

136     138 

-43   -96 

h 
109 

-83 

a 
2.      9390 
-2392 

b                c               d               e 

6908       9803      8803      8003 

-3580   -6199  -2514  -5720  - 

,/■ 
9433 
3384 

a 
3.       104 

-82    • 

bed 

103       203       502 

-59    -134    -179 

e            f 

639       503 

-99    -279 

9 
193 

-84 

a 

4.       938 
-246 

bed 
902       603     1090     : 
-593   -214   -836   - 

«            / 

1036       500 

990   -172 

9 

620 

-90 

34.   Oral  Exercises. 
Add: 

abcdefghij 
30*40     30     20     60     30     30     50     20     20 
20909020406030408060 

•  Dictate  to  the  class  thus :  20,  30.    The  pupils  add :  20,  50. 


ADDITION   AND  SUBTRACTION  — LESSON    B         61 
ADDITION  AND  SUBTRACTION  —  LESSON  B 

35.    Oral  Exercises. 

Md  as  indicated  in  Step  C,  p.  49. 


a 

6 

e 

d 

e 

/ 

9 

A 

i 

i 

fc 

I 

TO 

n 

9 

4 

8 

6 

3 

4 

3 

8 

4 

4 

3 

2 

4 

9 

3 

3 

3 

2 

4 

3 

9 

2 

3 

6 

9 

2 

8 

8 

4 

2 

6 

6 

2 

6 

4 

8 

3 

8 

2 

3 

2 

3 

3 

8 

4 

3 

3 

2 

4 

2 

9 

9 

8 

4 

6 

4 

4 

9 

4 

6 

6 

2 

3 

8 

2 

9 

4 

3 

2 

2 

9 

3 

5 

8 

4 

6 

3 

2 

8 

4 

6 

6 

2 

3 

36.    Oral  Problems. 

1.  One  year  is  —  months.     One  foot  is  —  inches. 

2.  A  boy  attends  school  9  months  in  the  year.    How 
many  months  of  vacation  does  he  have  ? 

3.  A  Hne  1  foot  long  is  divided  into  two  parts.    One 
part  is  3  inches  long.    The  other  part  is  —  inches  long. 

4.  Walter  has  4  apples  and  George  has  9  apples. 
George  has  how  many  more  apples  than  Walter  ? 

5.  Two  is  one  half  of  — .     One  half  of  6  is  — . 

6.  There  were  10  pints  of  milk  in  a  can.     Four 
pints  were  sold.    How  many  pints  of  milk  were  left  ? 

7.  A  boy  poured  2  pints  of  water  into  a  gallon  can. 
The  can  will  hold  —  more  pints. 

8.  Whatisthesumof  $6and$3?    Of  $3  and  $9? 

9.  A  boy  picked  9  boxes  of  berries.     He  sold  4  of 
them.     How  many  boxes  of  berries  had  he  left  ? 


62  ADDITION   AND  SUBTRACTION 

37.  Written  Exercises. 

a  h  c  d  e  f  g  hi 

423  229  736  322  763  722  333  227  343 

964  362  242  364  624  696  442  636  633 

629  432  633  429  232  242  328  852  433 

232  593  343  982  394  398  646  343  994 

242  864  943  262  323  443  612  964  663 

836  929  833  626  366  239  262  825  226 

38.  Written  Problems. 

Write  the  cost  of  the  second  purchase  under  the 
cost  of  the  first,  and  the  cost  of  the  third  purchase 
under  the  cost  of  the  second,  etc. 

1.  A  man  bought  a  horse  for  $43,  a  buggy  for 
$44,  a  harness  for  $23,  and  $26  worth  of  hay  and 
feed.     How  much  did  he  pay  for  all  ? 

2.  A  grocer  sold  228  pounds  of  sugar  on  Monday, 
333  pounds  on  Tuesday,  546  pounds  on  Wednesday, 
44  pounds  on  Thursday,  234  pounds  on  Friday,  95 
pounds  on  Saturday.  How  many  pounds  of  sugar 
did  he  sell  in  the  six  days  ? 

3.  What  is  the  sum  of  $3.94,  $6.33,  and  $8.96  ? 

4.  How  much  more  is  $9.30  than  $5.32? 


39.  Written  Exercises. 

a               b                 c 

1032      1204        8343 

-739     -224     -2319 

d 
5023 
-1124 

e 

9826 
-4620 

/ 

6600 
-3298 

ADDITION  AND  SUBTRACTION  — LESSON  B  63 

40.  Written  Problems. 

1.  A  farmer  had  93  sheep.  He  sold  34  sheep.  How 
many  sheep  did  he  have  left  ? 

2.  A  boy  was  flying  a  kite  with  a  string  400  feet 
long.  The  string  broke.  The  piece  in  the  boy's  hand 
was  220  feet  long.  What  was  the  length  of  the  piece 
on  the  kite  ? 

3.  There  are  48  pupils  in  Eoom  A  and  26  pupils 
in  Room  B.  How  many  more  pupils  are  there  in 
Room  A  than  in  Room  B  ? 

4.  There  are  48  pupils  in  Room  A,  26  pupils  in 
Room  B,  and  24  pupils  in  Room  C.  How  many 
pupils  are  there  in  the  three  rooms? 

5.  A  man  had  123  miles  to  travel.  He  traveled 
84  miles  the  first  day,  and  finished  his  journey  on  the 
next  day.     How  far  did  he  travel  the  second  day  ? 

41.  Find  the  amount  of  gain  or  loss  : 

1.  A  man  bought  a  horse  for  $120.  He  sold  it 
for  $88. 

2.  A  man  bought  land  at  $  95  an  acre.  He  sold  it 
at  $63  an  acre. 

3.  A  boy  bought  a  pony  for  $  26.    He  sold  it  for  $  50. 

4.  A  grocer  bought  flour  at  $.74  a  sack.  He  sold 
it  for  $1.00  a  sack. 

5.  A  horse  that  cost  $90  and  a  buggy  that  cost 
$40  were  both  sold  for  $145. 


64  ADDITION  AND  SUBTRACTION 


42.   1.    In  one  whole  there  are  —  halves. 

2.  In  one  whole  there  are  —  fourths. 

3.  In  one  whole  there  are  —  eighths. 

4.  In  one  half  there  are  —  eighths. 

5.  One  half  and  one  fourth  are  —  fourths. 

6.  One  half  and  one  eighth  are  —  eighths. 

7.  One  half  less  one  eighth  is  —  eighths. 

8.  One  half  less  two  eighths  is  —  eighths. 

9.  Four  eighths  are  —  times  one  eighth. 

10.  Three  fourths  are  —  eighths.  Six  eighths  are 
—  fourths. 

11.  A  boy  ate  one  fourth  of  a  pie.  How  much  of 
the  pie  was  left  ? 

12.  Three  boys  each  ate  one  fourth  of  a  pie.  There 
was  left  —  fourth  of  a  pie. 

13.  A  boy  had  one  fourth  of  a  pie.  He  cut  it  into 
two  equal  parts  to  share  it  with  another  boy.  Each 
boy  had  —  eighth  of  a  pie. 

14.  A  boy  paid  2  cents  for  one  half  dozen  apples. 
At  the  same  rate,  a  dozen  apples  would  cost  —  cents. 

15.  Using  lines,  show  one  half,  one  fourth,  and  one 
eighth. 


ADDITION  — LESSON   C  65 

ADDITION  — LESSON  C 
43.   1.   Memorize  the  following  : 

2  8  4  5  7- 

i        A         ^        A        A 

6  14  8  13  10 

2.  Give  a  number  story  suggested  by  each. 

3.  A  boy  bought  a  lead  pencil  for  5  cents  and  a 
tablet  for  8  cents.     How  much  did  he  pay  for  both  ? 

4.  Count  by  2's  to  10.     Count  by  3's  to  12. 

5.  Two  4's  are  — .     One  half  of  8  is  — . 

6.  In  going  home  from  a  store  a  boy  carried  4 
pounds  of  sugar  and  2  pounds  of  coffee.  What  was 
the  weight  of  both  packages  ? 

7.  A  girl  lives  4  blocks  from  the  schoolhouse. 
How  many  blocks  must  she  walk  in  going  home  and 
back  at  noon  ? 

8.  What  is  the  sum  of  $  8  and  $5?  Of  $6  and  $  8? 

9.  There  are  6  boys  and  8  girls  in  a  class.  How 
many  pupils  are  there  in  the  class  ? 

10.  Seven  years  and  —  years  are  ten  years. 

11.  George  has  8  rabbits  and  James  has  6  rabbits. 
Together  tliey  have  —  rabbits. 


44.    Read  the  following 

625,504         104,025 

$250.50 

$  1,050.75 

308,800         625,250 

1  305.25 

$  4,306.20 

700,040        300,406 

$670.07 

$  8,004.05 

1st  liK  Akitii— 5 

66  ADDITION   AND  SUBTRACTION 

45.  Study  Exercises. 

Study  as  indicated  in  Steps  A  and  B,  pp.  48  and  49. 

2  8  4  5  7 

4  6  4  8  3 

8  3  4  6  4 

5  7  4  8  2 

To  THE    Teacher.     Give   oral  drill  on  the  com- 
binations in  the  above  study.     See  p.  49. 

46.  Oral  Exercises. 

Add  as  indicated  in  Step  C,  p.  49. 
abcdefghij       k      I       m      n 

78284585778437 
57428864582873 
45744776858237 
84572354776673 
28457348587473 
46483746853437 


47.   Oral  Exercises. 

abed 

e 

/ 

9 

h 

i 

j 

20      80      40      50 

70 

40* 

30 

40 

60 

30 

40      60      40      80 

30 

42 

24 

58 

87 

95 

Dictate  problems,  thus :  A  boy  sold  40  papers  in 
the  morning  and  20  papers  in  the  evening.  How 
many  papers  did  he  sell  during  the  day  ? 

»  Dictate  thus :  42,  40.    The  pupils  add :  42,  82. 


SUBTRACTION  —  LESSON  C  67 


SUBTRACTION  — LESSON  C 

48. 

1.    Memorize  the  follovnng : 

10 

14         6       13         8       13       14 

6 

10 

-3 

7 

_6     -4     -5     -4     -8      -8 
8        2         8        4         5         6 

-2 
4 

-7 
3 

2.  Write  number  stories  suggested  by  each. 

3.  Three  boys  and  -^  boys  are  ten  boys. 

4.  Thirteen  days  are  8  days  and  —  days. 

5.  The  difference  between  $  14  and  $  6  is  $  — . 

6.  A  boy  has  $  3.     How  many  more  dollars  must 
he  have  in  order  to  pay  for  a  suit  of   clothes  that 

costs  $10? 

7.  What  must  be  added  to  4  to  make  6  ?    To  6  to 
make  14  ? 

8.  What  must  be  added  to  8  feet  to  make  14  feet  ? 

9.  Two  inches  and  —  inches  are  6  inches. 

10.  Ten  is  how  many  more  than  seven  ? 

11.  How  much  more  is  10  pounds  than  7  pounds  ? 

12.  Mary  worked  13  problems.  Alice  worked  8 
problems.  How  many  more  problems  did  Mary 
work  than  Alice  ? 

13.  A  girl  picked  14  flowers.  Eight  of  them  were 
roses.     The  rest  were  pinks.     She  picked  —  pinks. 

14.  What  is  5  less  than  13  ?  What  is  7  more 
than  3? 


68  ADDITION   AND  SUBTRACTION 

49.  Study  Exercises. 

10        14        6        13        8        6        13        14        10 

_3      _6    -4      -5    -4    -2      -8      -8       -7 

Study  the  above  exercises  until  you  can  give  the 
answers  without  hesitation. 

50.  Written  Exercises. 


a 

1.  104 

-26 

6 
146 
-64  - 

c     d 

638   1366 
-154  -542  - 

e 

863 
-418 

/ 

634 
-228 

1404 
-796 

a 
2.  1400 
-717 

b 

1033 

-248 

c     d 
6340  3604 
-1773  -566 

e 

1068 
-724 

/ 

4034 

-946 

fl' 

1000 

-627 

a                h                c                d 

3.  1403   1334   1432   9476 
-515  -842  -573  -870 

e 

6304 
-3766 

/ 
.  9030 
-3136 

a                     b                     c 
4.  $60.20   $82.83   $50.34 
-$27.87  -$13.14  -$17.46 

d 

$  10.23 
-$6.19 

e 

$  40.69 
-$2.30 

61.    Oral  Problems. 

1.  A  man   liad    13  horses.     He  sold  8  of  them. 
He  had  —  horses  left. 

2.  What  is  the  sum  of  6  inches  and  8  inches  ? 


ADDITION  AND  SUBTRACTION  — LESSON   C         69 

3.  A  girl  had  50  cents.  She  spent  20  cents  for 
some  ribbon.     She  had  —  cents  left. 

4.  How  many  pints  are  one  gallon  ? 

5.  If  each  family  takes  a  quart  of  milk,  a  gallon 
of  milk  will  supply  —  families. 

6.  There  are  24  boys  and  20  girls  in  a  school. 
There  are  — '-  pupils  in  the  school. 

7.  There  were  13  ducks  on  a  pond.  Nine  of  them 
flew  away.     There  were  —  ducks  left  on  the  pond. 

8.  A  boy  bought  a  tablet  for  10  cents  and  some 
marbles  for  8  cents.  How  much  money  did  he 
spend  ? 

9.  Ten  days  are  a  week  and  —  days. 

10.  What  is  the  answer  in  subtraction  called  ? 

11.  There  were  14  roses  on  a  bush.  Lottie  picked 
6  of  them.     There  were  —  roses  left  on  the  bush. 

12.  There  were  6  birds  in  a  tree.  Four  of  them 
flew  away.     There  were  —  birds  left  in  the  tree. 

13.  There  were  13  children  at  a  party.  Five  of 
them  were  boys.     How  many  girls  were  at  the  party? 

14.  Mary  has  8  cents.  Ethel  has  6  cents  more 
than  Mary.     Ethel  has  —  cents. 

15.  Harry  is  9  years  old.  Willie  is  3  years  older 
than  Harry.     Willie  is  —  years  old. 

16.  Alicfi  missed  3  words  in  her  spelling  lesson. 
She  had  12  words  to  spell.  She  spelled  —  words 
correctly. 


70  ADDITION   AND  SUBTRACTION 

52.   Oral  Exercises. 

Add  as  indicated  in  Step  C,  p.  49. 


a 

6 

c 

d 

e 

/ 

9 

A 

8 

i 

k 

; 

m 

n 

4 

5 

6 

8 

6 

8 

9 

4 

6 

8 

4 

3 

9 

7 

6 

4 

4 

9 

8 

4 

8 

3 

8 

9 

2 

4 

8 

5 

2 

2 

6 

3 

5 

9 

2 

4 

3 

4 

4 

6 

3 

4 

6 

9 

9 

7 

8 

4 

6 

2 

5 

6 

3 

9 

3 

8 

7 

5 

8 

9 

9 

6 

4 

6 

6 

4 

8 

2 

4 

2 

3 

8 

6 

4 

3 

3 

4 

4 

2 

9 

5 

2 

5 

4 

53.   Written  Problems. 

1.  A  farmer  had  400  acres  of  land.  He  sold  177 
acres.     How  many  acres  had  he  left? 

2.  A  farmer  had  27  acres  of  land.  He  bought  13 
acres.     How  many  acres  did  he  then  have  ? 

3.  Mary  has  68  cents.  Ethel  has  36  cents.  How 
much  money  have  they  both  ? 

4.  A  tailor  had  a  piece  of  cloth  containing  38  yards. 
He  cut  from  it  16  yards.  How  many  yards  were 
left  in  the  piece  ? 

5.  A  girl  had  90  cents.  She  spent  28  cents.  How 
much  money  did  she  have  left? 

6.  John  and  James  together  have  $1.00.  James 
has  $.34.     How  much  money  has  John? 

7.  A  horse  and  buggy  cost  $  200.  The  horse  cost 
$  120.     What  was  the  cost  of  the  buggy  ? 

8.  Fred  weighs  102  pounds.  Harry  weighs  79 
pounds.      How  much  heavier  is  Fred  tiian  Harry  ? 


ADDITION   AND  SUBTRACTION  —  LESSON  C  71 

54.    Oral  Exercises. 

1.  There  are  —  inches  in  one  foot. 

2.  There  are  —  feet  in  one  yard. 

3.  There  are  —  feet  in  3  yards. 

4.  Six  feet  are  —  yards.    Twelve  feet  are  —  yards. 

5.  Six  pints  are  —  quarts.      Eight  pints  are  — 
quarts. 

6.  Three  quarts  are  —  pints.     Four  quarts  are  — 
pints. 

7.  There  are  —  days  in  one  week.     One  year  is  — 
months. 

8.  Seven  feet  are  —  yards  and  —  foot. 

9.  Ten  days  are  —  week  and  —  days. 

10.   Name  the  months  of  the  year.     Which  have 
31  days? 


55.  Written 

Exercises. 

Add: 

a 

6 

c 

d 

e 

/ 

s 

1.  294 

252 

995 

839 

399 

963 

732 

367 

243 

852 

236 

416 

187 

202 

425 

263 

344 

434 

442 

832 

283 

934 

283 

569 

558 

353 

258 

304 

246 

265 

884 

622 

344 

466 

430 

628 

328 

679 

634 

685 

594 

809 

Subtract : 

a 

5 

C 

d 

e 

2.  $44.32 

1436  ft. 

9308  ft. 

$98.30   4344  yd. 

$23.39 

582  ft. 

5882  ft. 

$41.8( 

)      ; 

r58  yd. 

72  ADDITION  AND  SUBTRACTION 


A, 


B. 


C. 


66.  1.    Line  A  is  —  half  of  line  B. 

2.  Line  B  is  —  half  of  line  C. 

3.  Line  A  is  —  fourth  of  line  C. 

4.  Line  B  is  —  fourths  of  line  C. 

5.  Line  B  is  —  times  line  A. 

6.  Line  C  is  four  times  line  —  . 

7.  Line  C  is  two  times  line  —  . 

8.  If  line  A  represents  1  foot,  line  B  represents  — 
feet,  and  line  C  represents  —  feet. 

9.  If  line  A  represents  2  feet,  line  B  represents  — 
feet,  and  line  C  represents  —  feet. 

10.    If  line  A  represents  3  inches,  line  B  represents 
—  inches,  and  line  C  represents  —  inches. 

(«) : :       c) : : 

67.  1.    Divide  group  a  into  two  equal  groups. 

2.  Divide  group  h  into  four  equal  groups. 

3.  Two  dots  are  —  half  of  four  dots. 

4.  One  dot  is  —  fourth  of  four  dots. 

5.  If  each  dot  represents  $1,  in  group  a  there 
are  $  — . 

6.  $  2  is  —  half  of  $  4.     $  4  and  $  4  are  $  —  . 

7.  $4  is  —  times  $2.     One  half  of  $4  is  $  —  . 


ADDITION  — LESSON   D  73 

ADDITION— LESSON  D 

58.   1.    Memorize  the  follouy'mg  : 

2  6  5  7  5 

5  _7  3  _8  _5 

7  13  8  15  10 

2.  Write  a  number  story  suggested  by  each. 

3.  How  many  are  7  boys  and  6  boys  ? 

4.  How  many  are  7  girls  and  8  girls  ? 

5.  How  many  are  5  cents  and  5  cents  ? 

6.  What  is  the  sum  of  3  yards  and  5  yards  ? 

7.  What  is  the  sum  of  $  2  and  $  5  ? 

8.  Six  days  and  7  days  are  —  days. 

9.  Eight  weeks  and  7  weeks  are  —  weeks. 

10.  Seven  hours  are  5  hours  and  —  hours. 

11.  Ten  miles  are  5  miles  and  —  miles. 

12.  There  were  5  birds  in  a  tree.  Three  other  birds 
came  to  the  tree.    There  were  then  —  birds  in  the  tree. 

13.  A  boy  had  $6  in  his  bank.  He  put  $7  into 
the  bank.     He  then  had  $  —  in  the  bank. 

14.  Five  boys  were  camping  in  a  tent.  Three  other 
boys  joined  them.  There  were  then  —  boys  camping 
in  the  tent. 

16.  A  girl  spent  8  cents  for  some  paper  and  7  cents 
for  some  ribbon.     For  both  she  spent  —  cents. 

16.    Two  5's  are  — .     One  half  of  10  is  — . 


74 


ADDITION   AND   SUBTRACTION 


69.    Study  Exercises. 

Study  as  indicated  in  Steps  A  and  B,  pp.  48,  49. 
6  5  7  5 


To  THE  Teacher.     Give  oral  drill  on  the  above 
combinations.    See  p.  49. 

60.    Give  the  sums  of  the  following : 


a 
1.    28  and  7 

b 
23  and  5 

c 
52  and  5 

d 

77  and  6 

2.    37  and  6 

35  and  5 

67  and  8 

18  and  7 

3.    25  and  2 

17  and  6 

26  and  7 

22  and  5 

61.    Oral  Exercises. 

Add  as  indicated  in  Step  C,  p.  49. 


7n  n 

7  5 

5  6 

6  7 

7 


62.   Oral  Drill. 

Dictate  : 

abed 

e 

/ 

9 

h 

i 

J 

k 

20     60     80     50 

50 

•30 

70 

80 

90 

80 

40 

50     70     70     30 

50 

55 

67 

29 

48 

58 

95 

SUBTRACTION  — LESSON  D  75 

SUBTRACTION  — LESSON   D 

63.   1.    Memorize  the  foUoiving  : 

7       13         8       15       10         8       13         7       15 

5         65         8         5-3727 

2.  Give  a  number  story  suggested  by  each. 

3.  What  number  must  be  added  to  7  to  make  15  ? 

4.  What  number  must  be  added  to  2  to  make  7  ? 

5.  What  number  is  one  half  of  10  ? 

6.  Eight  is  3  more  than  — .     Two  and  —  are  7. 

7.  Seven  and  —  are  13.     Eight  and  —  are  15. 

8.  Eight  and  how  many  are  15  ?    Six  and  7  are  — . 

9.  What  number  is  5  less  than  7  ? 

10.  What  number  is  3  less  than  8  ? 

11.  Fifteen  is  7  more  than  — .     Ten  is  5  and  — . 

12.  A  girl  had  15  cents.  She  bought  a  tablet  for 
8  cents.     She  had  —  cents  left. 

13.  There  are  13  girls  and  7  boys  in  a  class.  How 
many  more  girls  than  boys  are  in  the  class  ? 

14.  There  are  8  quarts  of  water  in  one  pail  and  15 
quarts  of  water  in  another  pail.  There  are  —  quarts 
more  in  the  second  pail  than  in  the  first. 

15.  Ethel  has  8  books.  •  Mary  has  5  books.  Ethel 
has  —  books  more  than  Mary. 

16.  Harry  is  7  years  old.  His  brother  is  13  years 
old.     His  brother  is  —  years  older  than  Harry. 


76 


ADDITION   AND   SUBTRACTION 


64.    Study  Exercises. 


7       13         8       15       10         8 

13 

7      15 

_2     -7     -3      -7      -5      -5 

- 

■6     - 

-5     -8 

Study  the  above  until  you  can  give 

1  each  answer 

readily. 

65.   Written  Exercises. 

abed 

e 

/ 

1.     7385        8507        738o      1573 

1385 

7553 

-4628     -2752     -1647     -816 

- 

-728 

-4676 

2.     8393       9734       8143       6353 

7306 

9890 

-1634    -6186    -6098    -2665 

— 

1350 

-5397 

66.   Otal  Drill. 

1 

Dictate : 

a          b         c         d        e         f        9          ' 

k 

i 

./       fe 

23^  54     32     33     34     26     53      4 

7t 

27 

23     87 

52     45     34     75     64     34     47      48 

36 

59     56 

67.   Oral  Exercises. 

Add  as  indicated  in  Step  C,  p.  49. 

a       b       c       d       ('       f       (J       h        i 

./ 

k 

I     m     n 

6     3     5     7      2      7      5     7     4 

4 

7 

C     7     4 

349835229 

7 

3 

5     2     6 

4      74748888 

5 

9 

4     3     8 

3834657      2     5 

7 

4 

8     9     5 

672847     544 

3 

5 

2     2     2 

7356     9     8346 

3 

5 

4     2     8 

What  is  the  answer  in  subtraction  called  ? 

*  Add  :  52,  72,  75.  t  Add  :  48,  88,  95. 


ADDITION  AND  SUBTRACTION  — LESSON  D        77 

68.   Oral  Problems. 

1.  Harry  is  now   13    years  old.     In  how  many 
years  will  he  be  20  years  old  ? 

2.  When  Lottie  is  6  years  older,  she  will  be  14 
years  old.     How  old  is  Lottie  now  ? 

3.  After  spending  $  5  a  man  had  $  8  left.     He 
had  $  —  at  first. 

4.  A  pupil  added  two  numbers.     The  sum  was  15. 
One  of  the  numbers  was  8.    The  other  number  was  — ■. 

5.  A  boy  earns  $  13  a  month.     He  spends  $  6  a 
month.     The  boy  saves  $ —  each  month. 

6.  There  were  13  pupils  in  a  class.     Nine  of  them 
were  boys.     There  were  —  girls  in  the  class. 

7.  A  tablet  and  a  pencil  cost  9  cents.     The  tablet 
cost  5  cents.     The  pencil  cost  —  cents. 

8.  Ten  boys  are  how  many  more  than  6  boys? 

9.  Four  is  1  of  — .     What  is  l  of  10  feet  ? 

10.  Lucy  is   13  years  old.     Her  sister  is  6  years 
younger.     Her  sister  is  —  years  old. 

11.  A  boy  paid  15  cents  for  a  ball  and  bat.     The 
ball  cost  5  cents.     The  cost  of  the  bat  was  —  cents. 

12.  After  working  7  problems  a  girl  had  6  more  to 
work.     How  many  problems  had  she  at  first  ? 

•13.    There  are  14  eggs  in  two  nests.     In  one  of  the 
nests  there  are  8  eggs.     In  the  other  there  are  —  eggs. 
14.    Mary  is  now  9  years  old.     In  how  many  years 
will  she  be  13  years  old  ? 


78  ADDITION   AND  SUBTRACTION 

69.   1.    Draw  a  square  ;  divide  it  into  halves ;  mark 
each  half. 

2.  Draw  another   square;   divide  it  into  fourths; 
mark  each  fourth. 

3.  There  are  —  halves  in  a  square.     There  are  — 
fourths  in  a  square. 

4.  In  the  same  way  divide  a  square  into  eighths. 

5.  There  are  —  eighths  in  a  square. 

6.  How  many  fourths  make  one  half  ?     ^  =  f  •* 

7.  Fold  a  square  piece  of  paper  into  halves;  into 
fourths ;  into  eighths. 

8.  Show  6  eighths  of  a  square,     f  =  f . 

9.  Show  one  half,  one  fourth,  and  one  eighth  of  an 
oblong. 


70 

4 

.  Writ 

ten  E: 
3 

xercise 

'  Ik 

s. 

^ 

r 

/ 

i 

575 

797 

762 

659 

877 

386 

557 

647 

943 

687 

525 

522 

255 

535 

932 

786 

422 

427 

575 

694 

433 

762 

845 

567 

562 

593 

598 

356 

258 

844 

523 

944 

343 

555 

596 

697 

642 

722 

233 

476 

878 

738 

876 

282 

265 

965 

534 

206 

657 

256 

537 

757 

545 

358 

71 

.  Written  Exercises. 

« 

b 

c 

d 

e 

$610( 

)  $  5835 

$  23.34 

$  90.23 

9808  ft. 

— 

$2095  -$628  - 

-  $  8.76 

-$! 

).84  - 

-4472 

ft. 

*  Supply  the  value  of  x. 


ADDITION   AND  SUBTRACTION  — LESSON   D         79      ~- 

72.   Written  Problems. 

1.  A  man  bought  a  horse  for  $90.  He  sold  it 
for  $  60.     Find  the  gain  or  loss. 

2.  A  man  bought  a  cow  for  $  42.  He  sold  it  at 
a  gain  of  $  8.     Find  the  selling  price. 

3.  A  man  bought  a  lot  for  $600.  He  built  a 
house  that  cost  $  1200.  He  then  sold  the  property 
for  $  2300.     Find  the  gain  or  loss. 

4.  There  are  24  hours  in  one  day.  Find  the  num- 
ber of  hours  there  are  in  2  days. 

5.  There  are  52  weeks  in  one  year.  Find  the 
number  of  weeks  there  are  in  2  years. 

6.  A  man  owns  three  farms  of  320  acres  each.' 
Find  the  number  of  acres  in  the  three  farms. 

7.  A  wagon  loaded  with  coal  weighs  4280  pounds. 
The  wagon  weighs  1920  pounds.  What  is  the  weight 
of  the  coal  ? 

8.  What  is  the  sum  of  $  8.54,  $  7.86,  1 4.78,  and 

$6.85? 

9.  Find    the    difference    between    $106.75    and 

$43.48. 

10.  A  man  had  $900  in  a  bank.  He  drew  out. 
$300.  How  much  money  did  he  have  left  in  the 
bank  ? 

11.  A  man  borrowed   $73.     He  paid  back 
How -much  did  he  still  owe? 


80 


ADDITION   AND   SUBTRACTION 


73.  1.  A  pie  is  divided  equally  between  two  boys. 
Each  boy  gets of  the  pie. 

2.  Four  boys  divide  a  pie  equally  among  them- 
selves.    Each  boy  gets of  a  pie. 

3.  A  certain  number  of  boys  divide  a  pie  equally 
among  themselves.  Each  boy  gets  one  eighth  of  a 
pie.     There  are  —  boys. 

4.  ^  of  a  pie  is  equal  to  f  of  a  pie. 

5.  J  of  a  pie  and  |^  of  a  pie  are  f  of  a  pie. 


fi      14-1  =  ^        1-1-1   = 


4- 


/      \       •     •     •      • 

(a) 

^  ''   •  •  •  • 


(6) 


•  •     •     • 

•  •     •     • 


(«) 


•  •     •     • 

•  •     •     • 


74.  Divide  group  a  into  2  equal  groups;  group  h 
into  4  equal  groups ;  group  c  into  8  equal  groups. 

Let  each  dot  stand  for  a  pupil.  Fill  the  following 
blanks : 

1.  Four  pupils  are of  8  pupils. 

2.  Two  pupils  are ' of  8  pupils. 

3.  One  pupil  is of  8  pupils. 

4.  One  half  of  8  pupils  is  —  pupils. 

5.  One  fourth  of  8  pupils  is  —  pupils. 

6.  One  eighth  of  8  pupils  is  —  pupils. 


ADDITION  — LESSON   E  81 

ADDITION  — LESSON  E 
75.    1.    Memorize  the  following : 

7  6  7  9  4 

_1  _5  2  _9  _8 

16  12  9  18  12 

2.  Give  a  number  story  suggested  by  each. 

3.  What  is  the  sum  of  $  9  and  $  7  ? 

4.  Two  years  and  7  years  are  —  years. 

5.  Twelve  yards  are  6  yards  and  —  yards. 

6.  Twelve  yards  are  8  yards  and  —  yards. 

7.  One  half  of  18  miles  is  —  miles. 

8.  Six  quarts  and  6  quarts  are  —  quarts. 

9.  Six  is  \  of  — .     Three  is  |  of  — . 

10.  Seven  boys  and  2  boys  are  —  boys. 

11.  Six  eggs  and  6  eggs  are  —  dozen  eggs. 

12.  Eighteen  weeks  are  9  weeks  and  —  weeks. 

13.  Two  6's  are  — .     Two  9's  are  — . 

14.  Eighteen  weeks  are  —  times  9  weeks. 

15.  Two  books  are  what  part  of  4  books  ? 

16.  What  part  of  10  days  are  5  days  ? 

17.  What  part  of  a  gallon  is  one  quart  ? 

18.  Two  quarts  are  what  part  of  a  gallon  ? 

19.  Measure  15  feet  along  the  side  of  the  room, 
using  a  yard  stick.  Count  the  feet  by  3's  as  you 
measure. 

20.  Count  18  counters  by  2's  ;  by  3's. 

]ST  Bk  AiiiTir— (> 


82 


ADDITION  AND  SUBTRACTION 


76.   Study  Exercises. 

Study  as  indicated  in  Steps  A  and  B,  pp.  48,  49. 
7  6  7  9  4 

9  6  2  9  8 


To  THE  Teacher.      Give  oral  drill  on  the  com- 
binations in  the  above  study.     See  p.  49. 

77.    Oral  Exercises. 

Add  each  column  as  indicated  in  Step  C,  p.  49. 


a 

6 

c 

a 

e 

/ 

9 

A 

I 

i 

fc 

I 

m 

n 

4 

7 

4 

9 

7 

6 

7 

6 

9 

9 

9 

9 

6 

4 

9 

4 

9 

7 

6 

7 

4 

7 

7 

1 

1 

7 

9 

9 

7 

9 

7 

4 

7 

7 

9 

7 

6 

9 

9 

6 

1 

7 

6 

7 

4 

9 

7 

6 

7 

4 

7 

1 

1 

9 

7 

6 

7 

6 

9 

7 

4 

9 

8 

9 

2 

9 

9 

1 

1 

5 

9 

6 

9 

2 

8 

7 

4 

9 

7 

9 

7 

6 

8 

1 

78.    Find  the  amount  of  the  following  bill : 

Oakland,  Cal.,  June  5,  1905. 
Mr.  T.  H.  Crane, 

Bought  of  Horace  Mann  &  Co. 


2  doz.  eggs 

.        .     @$.20 

40 

2  lb.  ham  . 

.         .     @     .20 

40 

4  lb.  butter 

.         .     @    .25 

1 

00 

12  lb.  sugar 

.         .     @     .05 

60 

4  lb.  steak . 

.         .     @     .15 

60 

1  cabbage  . 

05 

SUBTRACTION  — LESSON  E  83 

SUBTRACTION  — LESSON   E 
79.    1.    Memorize  the  followmg : 


16 

9 

18 

12 

12 

16 

9 

12 

-9 

-2 

-9 

-6 

-8 

-7 

—  7 

-4 

77964928 

2.  Give  a  number  story  suggested  by  each. 

3.  What  must  be  added  to  9  to  make  16  ? 

4.  A  class  worked  16  examples  m  addition  and 
subtraction.  Seven  of  the  examples  were  in  subtrac- 
tion.    How  many  of  them  were  in  addition  ? 

5.  There  are  —  months  in  a  year. 

6.  If  a  boy  attends  school  8  months  each  year 
and  has  a  vacation  the  remaining  months,  he  has  — 
months'  vacation  each  year. 

7.  A  boy  bought  a  dozen  bananas.  He  ate  4  .of 
them.     He  had  —  bananas  left. 

8.  Nine  inches  and  —  inches  are  18  inches. 

9.  A  farmer  had  16  sheep.  He  sold  7  of  them. 
He  had  —  sheep  left. 

10.  A  grocer  sold  4  cans  of  corn  from  a-  box  con- 
taining a  dozen  cans.  There  were  —  cans  left  in  the 
box. 

11.  A  girl  bought  a  dozen  cookies.  She  gave  away 
all  but  4.     How  many  did  she  give  away  ? 

12.  On  Arbor  Day  the  pupils  planted  9  trees.  Two 
of  the  trees  died.     How  many  of  them  lived? 


84  ADDITION    AND  SUBTRACTION 


80. 

Study  Exercises. 

16 

12     9    12 

16 

9 

18 

12 

-7 

-6   -2   -8 

-9 

-7 

-9 

-4 

Study  the  above  until  you  can  give  the  results 
readily. 

81.  Written  Exercises. 

a                    h                    c                    d  e 

1.        9626           9828           9222           9826  5622 

-1859       -1859       -6738       -6877  -634 


2. 

9308 

9633 

1843 

8354 

8923 

-2529 

-1638 

-975 

-2866 

-  5237 

3. 

9386 

8926 

9540 

6345 

2873 

-6499 

-4180 

-6814 

-2378 

-916 

82.  Add :  Three  hundred  seventy  dollars  and 
twenty  cents,  forty -seven  dollars  and  forty-five 
cents,  one  hundred  four  dollars  and  eighty  cents, 
sixty-eight  dollars  and  forty-seven  cents,  one  dollar 
and  sixty-eight  cents,  three  hundred  fifty-two  dollars 
and  thirty-seven  cents,  sixty  dollars  and  eight  cents. 


83.  Oral  Exercises. 

a         b          c          d          e 

/ 

9 

h 

i 

J 

77   65   28   43   26 

84 

96 

47 

79 

57 

92   64   34   59   24 

64 

97 

83 

24 

40 

MULTIPLICATION  — LESSON   A  85 

MULTIPLICATION  — LESSON  A 

84.  1.    Find  the  sura  of  a  column  of  three  2's. 

2.  Find  the  sum  of  a  column  of  three  3's. 

3.  Two  3's  are  — .     Two  2's  are  — . 

4.  Three  2's  are  — .     Three  I's  are  — . 

5.  Two  O's  are  — .     Three  O's  are  — . 

6.  How  many  2's  are  there  in  4  ?     In  6  ? 

7.  How  many  3's  are  there  in  9  ?     In  6  ?    3 

8.  Two  3's  are  6  may  be  written  thus  :    x  2 

9.  Read  and  memorize :  ^ 

3  4  2  2  3  2 

x2  x^         x^         x3  x^  xjt 

~6         "^  4  6  9  8 

10.    The  sum  of  43  and  43  may  be  found  by  mul- 
tiplication, thus  : 

Model  :     43     Two  3's  are  6 ;  two  4's  are  8. 
X  2     The  answer  is  86. 


86 

85.    Multiply: 

^ 

a          b 

c 

d 

e 

/ 

9 

h 

1.     23       42 

30 

41 

14 

34 

24 

40 

x2      x2 

x2 

x2 

x2 

x2 

x2 

x2 

2.   234    403 

312 

231 

203 

123 

212 

120 

x2      x2 

x3 

x3 

x3 

x3 

x4 

x4 

The  answer  in  multiplication  is  called  the  product. 


86  ADDITION   AND  SUBTRACTION 

86.  1.   One  half  of  6  is  — .     One  half  of  10  is  — . 

2.  Four  is  J  of  — .     Five  is  ^  of  — . 

3.  Four  pints  are  —  quarts,  or gallon. 

4.  Four  quarts  are  —  pints.    Six  pints  are  —  quarts. 

5.  Two  yards  are  —  feet.    Nine  feet  are  —  yards. 

6.  Two  nickels  are  —  dimes.     Two  dimes  are  — 
cents. 

7.  One  dollar  is  —  cents.    One  dollar  is  —  dimes. 
One  half-dollar  is  —  cents,  or  —  dimes. 

8.  Six  dimes  are  —  cents.     There  are  —  half-dol- 
lars in  one  dollar. 

9.  There  are  —  quarter-dollars  in  one  dollar. 

10.  There  are  —  quarter-dollars  in  one  half-dollar. 

11.  There  are  —  nickels  in  one  quarter-dollar. 

12.  How  many  nickels  make  20  cents?   30  cents? 

13.  How  many  half-dollars  make  2  dollars  ? 

87.  Oral  Exercises. 

Add  each  column  as  indicated  in  Step  C,  p.  49. 


a 

b 

C 

d 

e 

/ 

9 

h 

I 

J 

k 

I 

m 

n 

7 

9 

8 

8 

3 

2 

3 

3 

4 

G 

5 

4 

2 

8 

6 

4 

2 

7 

9 

5 

7 

4 

6 

6 

5 

4 

3 

7 

5. 

2 

6 

9 

7 

9 

9 

7 

9 

6 

5 

4 

3 

9 

9 

4 

7 

7 

4 

6 

2 

9 

6 

C 

5 

4 

3 

9 

7 

8 

5 

6 

9 

6 

7 

6 

7 

6 

5 

4 

3 

7 

4 

7 

7 

2 

7 

8 

5 

8 

2 

6 

5 

4 

3 

9 

8 

9 

8 

4 

6 

4 

3 

5 

8 

6 

5 

4 

3 

4 

WRITTEN  PROBLEMS  87 

88.  Written  Problems. 

1.  A  man  bought  a  bicycle  for  $45  and  a  gun  for 
$38.  He  sold  both  for  $100.  Find  the  amount  of 
gain  or  loss. 

2.  A  boy  earns  $10  a  month  and  spends  $6. 
How  much  will  he  save  in  2  months? 

3.  What  must  be  added  to  $75  to  make  $120  ? 

4.  A  boy  had  65  cents.  How  much  money  had 
he  left  after  paying  25  cents  for  a  ticket  to  a  circus 
and  10  cents  for  some  popcorn  ? 

^  5.  A  girl  read  54  pages  of  a  book  on  Saturday 
and  25  pages  on  Sunday.  The  book  contained  102 
pages.  How  many  more  pages  has  she  to  read  to 
finish  the  book  ? 

6.  Mary  picked  16  quarts  of  berries  on  Monday, 
25  quarts  on  Tuesday,  17  quarts  on  Wednesday,  and 
8  quarts  on  Thursday.  How  many  quarts  did  she 
pick  in  all  ? 

7.  At  30  cents  each,  how  much  will  2  readers  cost  ? 

89.  Written  Exercises. 


a 

6 

c 

d 

e 

/ 

9 

h 

544 

737 

774 

269 

643 

309 

398 

232 

289 

446 

336 

441 

946 

721 

716 

796 

364 

655 

776 

269 

153 

809 

406 

749 

613 

737 

334 

441 

975 

941 

688 

959 

707 

983 

776 

434 

985 

969 

926 

474 

829 

795 

362 

239 

733 

259 

686 

656 

88  ADDITION   AND  SUBTRACTION 


90.  1.   One  whole  is  —  thirds. 

2.  One  whole  is  —  sixths.     One  half  is  —  sixths. 

3.  One  third  is  —  sixths.  Two  thirds  are  — 
sixths. 

4.  One  half  and  one  third  are  —  sixths. 

5.  One  half  of  a  pie  and  one  third  of  a  pie  are  — 
sixths  of  a  pie. 

6.  Two  thirds  and  one  half  are  —  sixths. 

7.  One  half  of  one  third  is  —  sixths. 

8.  Which  is  the  larger,  one  half  or  one  third  ? 
One  third  or  one  sixth  ?  One  half  or  three  sixths  ? 
One  half  or  two  sixths  ?    Two  thirds  or  three  sixths  ? 

9.  One  half  and  one  fourth  are  —  fourths. 

10.  The  ratio  of  2  to  4  is ;  of  :|  to  J  is ; 

of  |-  to  ^  is . 

11.  The  ratio  of  4  to  2  is  — ;  of  ^  to  ^  is  — ;  of 
•|  to  l  is  — . 

91.  Memorize: 

1.  One  half  and  one  fourth  are  three  fourths. 

2.  One  half  and  one  third  are  five  sixths. 


ADDITION  — LESSON  F  89 

ADDITION  — LESSON  F 

92.    1.    Memorize  the  following  : 

7  3  5  9  9 

_7  ^         J_  _1  _l 

14  7  12  11  10 

2.  Give  a  number  story  suggested  by  each. 

3.  What  is  the  sum  of  $  7  and  $  7  ? 

4.  How  many  days  are  there  in  2  weeks  ? 

5.  Two  7's  are  — .     One  half  of  14  is  — . 

6.  A  boy  spent  $  4  for  a  suit  of  clothes  and  $  3 
for  a  pair  of  shoes.     He  spent  $ —  in  all. 

7.  A  boy  had  10  words  to  spell.     He  missed  one 
word.     He  spelled  —  words  correctly. 

8.  Twelve  months  are  7  months  and  —  months. 

9.  How  much  more  is  the  sum  of  7  and  7  tlian 
the  sum  of  7  and  5  ? 

10.  The  sum  of  9  and  2  is  one  more  than  the  sum 
of  9  and  — . 

11.  If  one  sheep  costs  $  7,  what  will  be  the  cost  of  2 
sheep  ? 

12.  If  a  boy  had  12  oranges  and  gave  away  5  of 
them,  he  would  have  —  oranges  left. 

13.  Four  days  and  —  days  make  one  week. 

14.  A  ruler  12  inches  long  is  cut  into  2  pieces.  If 
one  piece  is  7  inches  long,  the  other  piece  is  —  inches 
long. 


90  ADDITION  AND  SUBTRACTION 

93.  Study  Exercises. 

Study  as  indicated  in  Steps  A  and  B,  pp.  48,  49. 
7  3  5  9  9 

7  4  7  2  1 


7 

2 

7 

1 

4 

5 

9 

7 

9 

3 

To  THE 

;  Teacher. 

Give  oral  drill. 

See  p.  49. 

94.   Oral  Exercises. 

Add  as 

in 

Step  C, 

p. 

49. 

a         b 

c 

d 

e 

/ 

9 

h 

i 

J 

k 

5       7 

9 

3 

9 

7 

9 

9 

5 

9 

4 

9       7 

9 

4 

9 

7 

5 

0 

3 

9 

9 

9       9 

5 

9 

7 

9 

3 

9 

7 

5 

9 

5       9 

3 

0 

5 

9 

4 

7 

3 

3 

5 

3       5 

7 

9 

9 

0 

9 

5 

7 

4 

4 

7       3 

4 

5 

9 

7 

2 

9 

3 

1 

0 

7       4 

3 

7 

2 

5 

9 

1 

4 

9 

3 

DIVISION  — LESSON  A 

95.    1.    Two  2's  are  — .     Two  3's  are  — .    Three 

2's   are — .     Three   3's   are   — .     Four   2's   are  — . 

Two  4's  are  — .  4 

2.  The  number  of  2's  in  8  may  be  shown  thus :  2)8 

3.  Bead  and  memorize  : 

1         1        1        A         1  1 

2)6            2)4            3)6            2)8            3)9  4)8 
The  answer  in  division  is  called  the  quotient. 


SUBTRACTION  — LESSON  F  91 

SUBTRACTION  — LESSON  F 

96.     1.    Memorize  the  following  : 

12       14       7       11       10       12       7       11       10 

^7      _7    -3      _2      -9      -5    -4      -9      -1 
574917329 

2.  Give  a  number  story  suggested  by  each. 

3.  A  girl  picked  7  boxes  of  cherries.  She  sold  all 
but  3  boxes.     How  many  boxes  did  she  sell  ? 

4.  Some  boys  bought  a  dozen  lemons.  They  used 
7  of  them  in  making  lemonade.  How  many  lemons 
had  they  left  ? 

5.  A  line  7  inches  long  is  —  inches  shorter  than 
a  foot  rule. 

6.  There  are  —  days  in  2  weeks.     Two  7's  are  — . 

7.  How  many  less  than  12  apples  are  7  apples  ? 

8.  How  many  less  than  11  weeks  are  9  weeks  ? 

9.  Seven  days  are  how  many  more  than  3  days  ? 

10.  Fourteen    inches    are   how   many   more   than 
7  inches  ? 

11.  How  much  longer  than  the  sum  of  4  inches 
and  3  inches  is  one  foot  ? 

12.  How  much  longer  than  the  sum. of  2  inches 
and  9  inches  is  one  foot  ? 

13.  How  many  hours  are  there  from  5  o'clock  to 
12  o'clock? 


92  ADDITION   AND  SUBTRACTION 

97.  Study  Exercises. 

12         7       11       10       14       12       11         7       10 

-7     _3     -9     -1     -7      -5      ^2     -4      -9 

Study  the  above  until  you  can  give  the  results 
without  hesitation. 

98.  Written  Exercises. 


1. 


a 

7474 
-3737 

b 
7272 
-2727 

C 

7242 
-3465 

d 
2010 
-1999 

e 

7111 
-3182 

a 
8401 
-2749 

b 
9123 
-3145 

c 
9445 

-5677 

d 
7452 
-4567 

e 

3572 

-816 

99.  Study  Exercises. 

2)6  2)4  3)6  2)8  3)9  4)8 

Study  the  above  until  you  can  give  the  answers 

readily. 

The  number  of  2's  there  are  in  64  may.  be  found 

32 
thus :  ^T-^      There  are  three  2's  in  6,  and  two  2's 

in  4.     There  are  thirty-two  2's  in  64. 

100.  Divide-. 

a  b  c  d  e 

1.   2)46  2)62  2)80  2)64  3)63 


2.  2)6420      2)2604      2)4026       2)4602      2)2064 

3.  3)3690      3)6309      3)'9603       3p»96      3)"63()9 


ORAL  PROBLEMS  98 

101.   Oral  Problems. 

1.  How  many  pints  are  there  in  3  quarts  ? 

2.  How  many  yards  are  there  in  9  feet  ? 

3.  Three  dollars  are  how  many  half-dollars  ? 

4.  Eight  pints  are  how  many  quarts  ? 

5.  Two  yards  are  —  feet.     Two  gallons  are  — 
quarts. 

6.  A  half-dollar  is  —  dimes  ;  —  nickels ;  —  cents. 

7.  Eighteen  cents  are  —  dime  and  —  cents. 

8.  Six  quarts  are  —  gallon  and  —  quarts. 

9.  The  number  of  2  cents  there  are  in  6  cents 
may  be  shown  thus  :        _§_ 

10.  Read  and  give  quotients  : 

2^)47      3^)6?       4^)8?       2^)87      $3)p       $2)p 

11.  Read  and  divide : 


3ft.)66ft.  $4)$48  $2)$80  2  in.)60  in. 

102.    Oral  Exercises. 

Add  each  column  as  indicated  in  Step  C,  p.  49. 


a 

6 

C 

d 

e 

/ 

g 

/« 

I 

J 

k 

z 

m 

W 

4 

9 

7 

6 

3 

2 

8 

8 

7 

4 

5 

8 

6 

7 

6 

2 

9 

8 

8 

6 

6 

7 

4 

9 

6 

4 

4 

8 

7 

6 

9 

7 

7 

9 

2 

8 

7 

6 

8 

5 

6 

7 

6 

4 

8 

8 

6 

7 

8 

6 

4 

7 

6 

9 

8 

9 

8 

6 

7 

7 

8 

4 

7 

7 

6 

1 

7 

2 

6 

7 

5 

7 

2 

8 

7 

4 

3 

7 

3 

4 

4 

6 

9 

9 

7 

9 

5 

5 

7 

8 

4 

2 

5 

5 

9 

6 

9 

3 

94  ADDITION   AND  SUBTRACTION 

103.    1.    We   can   find   one   half    of    8   books   by 
separating  the  books  into  —  equal  groups. 

2.  Show  one  half  of  6  books ;  of  8  books. 

3.  One  half  of  8  books  is  —  books.     This  may  be 
shown  thus  :    4  books 

2)8  books 

4.  Read  and  give  quotients : 


2)4  books     3)6  books     2)$ 8     2)$  6      3)9  ft. 

5.   Read  and  divide  : 
abed 


2)$  286      2)$ 402      2)$ 840      2)608  ft.      3)360  da. 

6.    Find  1  of  $460.     Find  1  of  $390. 

-  7.    Find  i  of  $408.     Find  l  of  680  pounds. 

8.   A  man  had  $84.     He  spent  one  half  of  it  for  a 
wagon.     How  much  did  the  wagon  cost  him  ? 

104.   Written  Exercises. 


a 

b 

C 

d 

e 

/ 

9 

h 

547 

923 

897 

246 

875 

141 

646 

574 

656 

759 

265 

463 

336 

676 

246 

306 

878 

386 

682 

675 

474 

519 

346 

759 

697 

779 

376 

978 

385 

434 

346 

686 

874 

525 

449 

926 

673 

608 

346 

853 

382 

467 

772 

437 

469 

187 

346 

418 

492 

487 

839 

299 

843 

515 

346 

707 

WRITTEN  PROBLEMS  95 

105.    Written  Problems. 

1.  After  selling  47  sheep  a  farmer  had  left  38 
sheep.     How  many  sheep  had  he  at  first  ? 

2.  A  farmer  had  32  cows.  He  bought  29  more 
cows.     How  many  cows  had  he  then  ? 

3.  Of  a  school  of  436  pupils,  169  are  boys.  How 
many  girls  are  there  in  the  school  ? 

4.  Find  the  cost  of  3  cows  at  $  23  each. 

5.  There  are  55  pupils  in  the  First  Grade,  46 
pupils  in  the  Second,  and  37  pupils  in  the  Third. 
How  many  pupils  are  there  in  the  three  grades  ? 

6.  There  are  35  girls  and  18  boys  in  a  school. 
How  many  more  girls  than  boys  are  there  in  the 
school  ? 

7.  A  boy  had  48  marbles.  He  sold  one  fourth  of 
them.  How  many  did  he  sell  ?  How  many  marbles 
did  he  have  left  ? 

8.  A  grocer  bought  flour  at  89  cents  a  sack  and 
sold  it  at  $  1.00  a  sack.  How  much  did  he  make  on 
each  sack  ? 

9.  A  horse  that  cost  $  86  was  sold  at  a  gain  of 
$  18.     Find  the  selling  price. 


106. 

Solve : 

a 

6 

c 

d 

e 

/ 

$234 

$403 

$320 

432  ft. 

201  ft. 

233  yd. 

x2 

x2 

x3 

x2 

x4 

x3 

ft.  ft.  yd, 


96  ADDITIOISr  AND  SUBTRACTION 

9     •     •  ,  •     •     •  ••• 

(a)  (b)  (c) 

107.  Divide  group  a  into  two  equal  parts;  group 
h  into  three  equal  parts ;  group  c  into  six  equal  parts. 
Let  each  dot  represent  a  pupil. 

1.  —  pupils  are  ^  of  6  pupils. 

2.  —  pupils  are  J  of  6  pupils. 

3.  —  pupil  is  ^  of  6  pupils. 

4.  Six  pupils  are  —  times  3  pupils. 

5.  Six  pupils  are  —  times  2  pupils. 

6.  How  many  2  pupils  are  there  in  6  pupils  ? 

7.  How  many  3  pupils  are  there  in  6  pupils  ? 

8.  Six  pupils  are  how  many  times  2  pupils  ? 

9.  Six  pupils  are  how  many  times  3  pupils  ? 

10.  One  half  of  6  pupils  is  —  more  pupil  than 
one  third  of  6  pupils. 

11.  The  ratio  of  3  pupils  to  6  pupils  is  —  — ;  of 
2  pupils  to  6  pupils  is . 

12.  The  ratio  of  6  pupils  to  3  pupils  is  — ;  of  6 
pupils  to  2  pupils  is  — ;  of  6  pupils  to  one  pupil  is  — . 

13.  The  difference  between  J  of  6  pupils  and  -^  of 
6  pupils  is  —  pupil. 

108.  Add: 


H 

5* 

9^ 

2i 

H 

H 

ii 

H 

2i 

H 

ADDITION  — LESSON  G  97 

ADDITION  — LESSON  G 
109.    1.    Memorize  the  following  : 

8  9  6  7  9 

_8  _6  _5  1  _1 

16  15  11  8  17 

2.  Give  a  number  story  suggested  by  each. 

3.  What  is  the  sum  of  9  and  8  ? 

4.  What  is  the  sum  of  two  8's  ?  ■ 

5.  A  boy  has  9  marbles  in  one  pocket,  and  6  in 
the  other.  How  many  marbles  has  he  in  both 
pockets  ? 

6.  Seventeen  is  —  more  than  8. 

7.  Two  8's  are  16.     One  half  of  16  is  — . 

8.  What  is  the  ratio  of  16  to  8  ?     Of  8  to  16  ? 

9.  Fred  has  6  pigeons.  Walter  has  5  pigeons 
more  than  Fred.     How  many  pigeons  has  Walter  ? 

10.  A  piece  5  ft.  long  was  sawed  from  a  board  11  ft. 
long.     How  long  was  the  part  that  remained  ? 

11.  A  post  8  ft.  long  is  1  ft.  below  ground.  How 
long  is  the  part  above  ground  ? 

12.  A  boy  delivers  6  quarts  of  milk  each  morning 
and  5  quarts  each  evening.  How  many  quarts  does 
he  deliver  each  day? 

13.  From  a  board  16  ft.  long  a  piece  8  ft.  long  is 
cut.     How  long  is  the  part  remaining  ? 

1st  ]\k  Aimth  _7 


98  ADDITION   AND  SUBTRACTION 

110.  Study  Exercises. 

Study  as  indicated  in  Steps  A  and  B,  pp.  48,  49. 

8  9  6  7  9 

8  6  5  18 

8  18  5  6 

9  7  8  6  9 

To  THE  Teacher.     Dictate  for  oral  addition  the 
combinations  studied  in  the  above  exercises.    See  p»  49. 

111.  Oral  Exercises. 

Add  each  column  as  indicated  in  Step  C,  p.  49. 


a 

6 

C 

d 

e 

/ 

g 

h 

i 

j 

k 

I 

m 

n 

9 

1 

6 

6 

5 

6 

1 

8 

9 

8 

9 

9 

9 

5 

7 

9 

9 

9 

8 

9 

9 

1 

7 

7 

8 

1 

1 

8 

6 

7 

8 

8 

7 

8 

7 

9 

5 

5 

1 

9 

9 

7 

9 

6 

1 

7 

6 

1 

5 

7 

1 

8 

9 

1 

1 

5 

8 

9 

9 

5 

6 

8 

8 

6 

0 

7 

0 

9 

6 

8 

8 

6 

8 

6 

9 

9 

8 

5 

9 

1 

8 

8 

9 

8 

113.   Write  in  a  column  and  add: 

1,  138.67,  $.88,  $67.46,  $.89,  $69.34. 

2.  $85.89,  $.70,  $8.05,  $67.96,  $9.77. 

113.    Write  and  solve : 

1.  $24.93 -$8.15.  4.  $30.00 -$6.44, 

2.  $104.50 -$7.15.  5.  $90 -$17.50. 

3.  $70.42-$5.79.  6.  $85.46-118. 


15 

11 

17 

8 

-9 

-6 

-9 

-1 

6 

5 

8 

7 

s 

SUBTRACTION  — LESSON   G  99 

SUBTRACTION  — LESSON  G 

114.  1.    Memorize  the  following : 

15       11       17         8       16 

-6     -5     -8     -7     -8 
9         6         9         18 

2.  Give  a  number  story  suggested  by  each. 

3.  A  man  earns  $15  a  week  and  spends  $6  a 
week.     He  saves  $ —  each  week. 

4.  A  post  11  feet  long  stands  in  a  hole  5  feet  deep. 
How  much  of  the  post  is  above  ground  ? 

5.  A  milkman  sold  8  quarts  of  milk  from  a  can 
containing  17  quarts.  How  many  quarts  remained 
in  the  can  ? 

6.  A  grocer  sold  15  lb.  of  sugar  in  two  packages. 
If  one  of  the  packages  weighed  9  lb.,  how  much 
did  the  other  weigh  ? 

7.  Mary  is  8  years  old  and  her  sister  is  17  years 
old.     Mary  is  —  years  younger  than  her  sister. 

8.  Ethel  went  to  visit  Lottie  on  the  ninth  of  June 
and  stayed  until  the  fifteenth  of  June.  How  long 
was  her  visit  ? 

115.  Multiply: 

ah  c  d  e  f 

423   3132   2012   4023   1022   3012 
2     3     4     2     4     3 


100  ADDITION   AND   SUBTRACTION 

116.    Study  Exercises. 
15       11         8       17       16       15        8       11        17 

_9_5_^      _8      zl     Z^     zl     IL^      zl 

Study  the  above  until  you  can  give  the  results 
without  hesitation. 


117.  Written  Exercises. 

a                 b                 c 

d 

e 

/ 

1.  8515   8717   1755 

8655 

1556 

1777 

-849  -6859  -886 

-7759 

-858 

-878 

2.  1715 

5365 

9537 

9608 

7474 

1423 

-849 

- 1578 

-2769 

-6809 

-1837 

-348 

3.  9847 

6328 

8456 

9167 

3135 

8636 

-2038 

- 1885 

-3778 

-5569 

-439 

-2747 

MULTIPLICATION  — LESSON  B 
118.    Show  by  addition  and  multiplication : 

1.  The  sum  of  two  5's ;  of  four  3's ;  of  three  4's. 

2.  The  sum  of  five  2's  :  of  two  6's  ;  of  six  2's. 


119.    Bead  and  memorize : 
1.     5  4  2  3 

x2       x3       x5      x5 

To      12      lo     T5 


6 

3 

2 

•  5 

x2 

x4 

x6 

x3 

12 

12 

12 

15 

2.    What  is  the  answer  in  multiplication  called? 


MULTIPLlCATIOfN-'-.LKSO:^^'  h  101 


120. 

Study  Exercises. 

5 

4          2          5 

6 

3 

2 

x2 

x3        x5        x3 

x2 

x4 

x6 

x5 


Study  the  above  until  you  can  give  the  results 
without  hesitation. 

Give  the  products  in  the  above  exercises  from 
right  to  left,  adding  1,  2,  and  3  to  each  product,  thus : 
15,  16  ;  12,  13  ;  12,  13  ;  etc. 

121.   Written  Exercises. 

Model  for  Exercise  a:    465         Carry  in    multi- 

2     plication  as  in  ad- 

930     dition. 


o 

5 

c 

d 

e 

/ 

(1 

7i 

1.    405 

365 

260 

345 

250 

123 

203 

332 

x2 

x2 

x2 

x3 

x3 

x4 

x4 

x-4 

2.    210 

122 

201 

231 

102 

405 

604 

213 

X  5 

X  5 

x6 

x5 

x6 

x3 

x2 
403 

x4 

3.    345 

234 

222 

320 

1.32 

305 

413 

x2 

x3 

x6 

x4 

x5 

x2 

x2 

x3 

122.    Oral  Exercises. 

Add: 

0             b 

C 

d 

e 

/ 

0 

h 

i 

70       69 

75 

53 

58 

43 

36 

07 

25 

70   ^-  61 

86 

45 

64 

59 

94 

83 

76 

102  ADDITION   ANO   SUBTRACTION 

DIVISION  — LESSON  B 

123.  1.  Count  by  3's  to  18 ;  by  2's  to  24 ;  by  4's 
to  20 ;  by  5's  to  30. 

2.  Arrange  6  books  to  show  that  2  books  are  ^  of 
6  books. 

3.  One  third  of  6  books  is  —  books.  Two  thirds 
of  6  books  are  —  books. 

4.  Draw  oblongs  to  represent  9  books.  One  third 
of  9  books  is  —  books.  Two  thirds  of  9  books  are 
—  books. 

5.  Show  J  of  12  squares.     Show  |  of  12  squares. 

6.  Three  fourths  of  12  squares  are  —  squares. 

7.  Show  1  of  12  circles.     Show  ^  of  12  circles. 

124.  Bead  and  memorize  : 
43352562 


3)12     5)15     4)12     3)15     6)12     2)10     2)12     5)10 

125.    Oral  Exercises. 

Add  as  indicated  in  Step  C,  p.  49. 


a 

h 

C 

d 

e 

/ 

9 

h 

i 

J 

A; 

I 

m 

n 

3 

6 

4 

5 

4 

7 

3 

9 

7 

5 

9 

8 

4 

3 

7 

8 

7 

6 

G 

G 

8 

7 

8 

7 

6 

9 

8 

5 

9 

7 

9 

8 

2 

8 

2 

8 

2 

6 

7 

1 

9 

f> 

7 

8 

8 

5 

3 

i 

7 

9 

5 

9 

6 

9 

7 

8 

5. 

6 

3 

G 

8 

9 

4 

7 

6 

9 

9 

1 

4 

7 

7 

5 

G 

8 

9 

7 

6 

4 

8 

7 

5 

9 

6 

G 

9 

8 

7 

9 

6 

2 

8 

9 

4 

5 

C 

8 

6 

4 

DIVISION  — LESSON  B  103 

126.  Study  Exercises. 

3)12     5)T5     4jTI    3)15     6)T2     2)l0     2)T2     5)T0 

Study  the  above  until  you  can  give  the  quotients 
without  hesitation. 

1.  2)12     This  means :  How  many  2'5  are  there  in 
12?     Or,    What  is  ^  of  12  2     The  answer  is  — . 

2.  2)p2     This  means:    What  is  J  of  $12?     The 
answer  is  — . 

3.  $2)$  12     This  means:  Hoiv  many  $2  are  there 
in  $12?     The  answer  is  — . 

127.  Read  and  find  quotients  : 

a  h  c  d  e 


1.  2)2648     4)8048  4)1248     2)1046  $3)$  1296 

2.  6)1206     5)T050  5)T005     4)1^08  3)6012  ft. 

128.   Written  Exercises. 

a  h           c            d  6           /            g  h            i 

799  999  788  899  949  997  978  889  998 

617  899  476  937  994  556  292  789  871 

696  991  768  946  794  958  926  875  358 

949  946  489  479  893  779  374  965  663 

978  999  987  558  794  497  898  899  994 

538  426  259  538  369  838  366  429  439 


129.    Solve: 

a 

6 

C 

d 

$84.93 

$43.21 

$87.53 

$97.34 

-$17.16 

-$14.52 

-$48.79 

-$17.36 

104 


AUDITION   AND   SUBTRACTION 


Halves 


Thirds 


Fourths 


Sixths 


Eighths 


130.    Show  the  truth  of  each  statement  by  folding 
or  cutting  paper. 

1.  ^  of  a  pie  is  more  than  ^  of  a  pie. 

2.  ^  of  a  pie  is  less  than  ^  of  a  pie. 

3.  ^  of  a  pie  + 1-  of  a  pie  is  the  same  as  -J  of  a  pie. 

4.  ^  of  a  pie  +  ^  of  a  pie  is  more  than  -|^  of  a  pie. 

5.  f  of  a  pie  +  i^  of  a  pie  is  ^  of  a  pie  less  than  a 
whole  pie. 

6.  I  of  a  pie  +  J  of  a  pie  is  ^  of  a  pie  more  than  a 
whole  pie. 

7.  1  of  a  pie  +  ^  of  a  pie  is  ^  of  a  pie  less  than  a 
whole  pie. 

8.  I  of  a  pie  is  J-  of  a  pie  less  than  ^  of  a  pie. 

9.  I  of  a  pie  +  J  of  a  pie  is  |-  of  a  pie  more  than 
^  of  a  pie. 

10.    If  we  cut  :|-  of  a  pie  from  |  of  a  pie,  there 
will  remain  ^  of  a  pie. 


131. 

Memorize : 

a 

b 

c 

d 

h 

t 

1 

f 

i 

-i 

+i 

+  i 

i 

i 

5 
8 

i=H 

+i 


ADDITIOX  — LESSON   H  105 


ADDITION  — LESSON  H 

132.    1. 

Memorize  the  following  : 

1 

5              7            2 

8 

8 
9 

9              4            1 
14            11            3 

3 
11 

2.  Give  a  number  story  suggested  by  each. 

3.  What  two  combinations  in  this  lesson  give  11 
as  a  sum  ? 

4.  What  must  be  added  to  $5  to  make  $14  ? 

5.  Name  another  combination  whose  sum  is  14. 

6.  The  sum  of  9  and  9  is  18  ;  of  9  and  8  is  17  ;  of 
9  and  7  is  16.  When  a  number  is  added  to  9,  the 
sum  ends  in  a  figure  one  less  than  that  added  to  9. 

7.  A  girl  has  4  books  of  poems  and  7  story  books. 
How  many  books  has  she  in  all  ? 

133.  Oral  Exercises. 

The  sign  =  between  two  quantities  shows  that 
they  are  equal  in  amount. 

4  +  5  =  9.  This  means  that  the  sum  of  4  and  5  is 
equal  to  9. 

4  +  5  =  6  +  3.  This  means  that  the  sum  of  4  and 
5  is  equal  to  the  sum  of  6  and  3. 

134.  Supply  the  number  that  should  stand  in  place 

of  X. 

1.  7  +  4  =  3+^'.     3.  9  +  7  =  8  +  ;r.     5.  7  +  8  =  6 +  a;. 

2.  9  +  5  =  8  +  ^:.     4.  6  +  5  =  4  +  a:.     6.  5  +  9  =  7+ic. 


106  ADDITION   AND  SUBTRACTION 

135.  Study  Exercises. 

Study  as  indicated  in  Steps  A  and  B,  pp.  48,  49. 

15  7  2  8 

8  9  4  13 

3  19  4  8 

8  2  5  7  1 

To  THE  Teacher.     Dictate  the  combinations  in- 
volved in  the  above  study.     See  p.  49. 

136.  Oral  Exercises. 

Add  as  indicated  in  Step  C,  p.  49. 
a        b        c        d        e         f       g        hi       j        k        I       m 

8  7  578288  9  8822 
2-5  85278712787 
7888852598  5  29 
5728287812881 
1582739893415 
893148   5358729 


137.   Drill  columns.     A  drill  column  is  one  in  which  a  g 

combination  occurs  several  times.   To  make  a  drill  column  n 

'for  8  and  7:  Write  the  combination  at  the  foot  of  the  ^ 
column.     The  sum  is  15.     Place  in  the  column  a  number 

that  will  increase  the  sum  to  either  18  or  17.     This  number  ^ 

is  either  3  or  2.    Either  can  be  used.    If  2  is  taken,  the  sum  8 

is  increased  to  17.     Then  place  8  in  the  column.     The  sum,  2 

is  25.     Again  add  either  3  or  2,  and  continue  as  above.  n 

Write  a  drill  column  for  6  and  7;  for  9  and  6;  for  8 
and  0;  for, 9  and  7. 


8 


SUBTRACTION  — LESSON  H  107 


SUBTRACTION  —  LESSON 

H 

138.    1. 

Memorize  thefolloioing  : 

9      14 

11        9        3      11      14 

11 

11 

3 

-1    -5 

-4    -8    -2     -8     -9 

-7 

-3 

-1 

8        9 

7       113        5 

4 

8 

2 

2.  Give  a  number  story  suggested  by  each. 

3.  A  board  11  ft.  long  will  make  two  shelves,  one 
4  ft.  long  and  the  other  —  ft.  long. 

4.  How  many  days  are  there  from  April  3  to 
April  11? 

5.  There  were  11  marbles  in  a  ring.  Frank  shot 
3  of  them  out  of  the  ring.  There  were  —  marbles 
left  in  the  ring. 

6.  A  farmer  sold  14  sacks  of  grain.  Five  were 
wheat,  and  the  rest  were  oats.  He  sold  —  sacks  of 
oats. 

Draw  a  diagram  to  show  the  places  mentioned  in 
each  of  the  following  problems  : 

7.  Harry's  home  is  1  mile  north  of  the  school- 
house,  and  Willie's  home  is  2  miles  south  of  the  school- 
house.     How  far  apart  do  they  live  ? 

8.  Mary's  home  is  4  blocks  east  of  the  school- 
house,  and  Edna's  home  is  7  blocks  west  of  the 
school  house.     How  far  apart  do  they  live  ? 

9.  Fred  lives  9  miles  west  of  the  city,  and  James 
lives  14  miles  west  of  the  city.  How  far  apart  do 
they  live  ? 


108  ADDITION   AND   SUBTRACTION 

139.    Study  Exercises. 

9       14       11       9       3       11      14      11      11       8 

-1     -5     zl  ZL^  zl     -8     -9     -7     -3   -1 

Study  the  above  until  you  can  give  the  results 
without  hesitation. 


140.   Written  Exercises. 

a 

1.   3841 
-1687 

6 

9341 

-944 

C 

3114 

-275 

d 
3411 
-2463 

e 
9141 

-7398 

/ 
9319 
-1041 

2.    6056 
-3968 

9327 
-8469 

2526 

-783 

8436 
-4759 

5143 
-2436 

9357 
-6468 

3.    9418 
-4476 

9473 
-6627 

9365 
-7467 

9785 
-6489 

3368 
-2579 

7991 
-4898 

141. 

Oral  Exercises. 

Add  each  columr 

1  as 

indicated  in 

Step  C, 

P' 

i9. 

a      b 

c 

d 

e 

/ 

0 

h 

i 

j 

A- 

/ 

m 

« 

6     8 

9 

7 

5 

9 

9 

4 

3 

9 

5 

4 

2 

9 

9     9 

5 

9 

6 

5 

7 

0 

5 

5 

4 

3 

8 

7 

4     1 

5 

1 

4 

5 

8 

6 

5 

•5 

6 

7 

2 

3 

6     6 

7 

4 

8 

6 

7 

9 

8 

9 

7 

9 

8 

9 

5     4 

S 

6 

2 

4 

3 

7 

2 

1 

3 

1 

2 

1 

8     9 

5 

4 

6 

5 

7 

3 

8 

5 

7 

9 

8 

i 

9     6 

7 

9 

8 

6 

7 

8 

5 

9 

4 

3 

8 

9 

ORAL  PROBLEMS  109 

142.  Oral  Problems. 

1.  One  dollar  is  —  cents  ;  —  nickels ;  - —  dimes  ; 

—  half-dollars. 

2.  A  half-dollar  is  —  cents;  —  nickels;  —  dimes; 

—  quarter-dollars. 

3.  A  quarter-dollar  is  —  cents;  —  nickels;  2 
dimes  and  —  cents.    ^ 

4.  If  Edna  buys  a  box  of  berries  for  15  cents  and 
gives  the  clerk  a  2o-cent  piece,  how  much  change  will 
she  receive  ? 

5.  Mabel  buys  35  cents'  worth  of  sugar  and  gives 
the  clerk  a  half-dollar.  The  clerk  counts  the  change 
as  he  gives  it  to  Mabel.  He  begins  with  the  cost  of 
the  sugar  and  says,  35  and  5  are  40,  and  10  are  50, 
as  he  gives  her  a  nickel  and  a  dime. 

6.  Ethel  bought  30  cents'  worth  of  ribbon  and 
handed  the  dealer  a  half-dollar.     Count  the  change. 

143.  Have  the  pupils  take  turns  at  ^'  keeping  store." 
Supply  them  with  paper  coins  (or  better,  with  real 
coins),  and  have  them  make  purchases  and  count  the 
change. 

Make  change  for : 

1.  40  cents  out  of  $1.00.  5.  30  cents  out  of  $1.00. 

2.  60  cents  out  of  $1.00.  6.  15  cents  out  of  $5.00. 

3.  $1.25outof  $5.00!  7.  $3.50  out  of  $10.00. 

4.  $2.25  out  of  $5.00.  8.  $4.25  out  of  $5.00. 


110  ADDITION   AND  SUBTRACTION 

144.    Oral  Problems. 

1.  A  boy  paid  50^  for  a  baseball  and  30^  for  a 
glove.     How  much  did  he  pay  for  both  ? 

2.  Harry  and  James  picked  two  boxes  of  apples 
and  sold  them  at  60  ^  a  box.  How  much  did  they  get 
for  both  boxes? 

3.  If  there  are  24  boys  and  32  girls  in  the  school, 
how  many  children  are  there  in  the  school  ? 

4.  Six  boys  bought  a  dozen  bananas  and  shared 
them  equally.  How  many  bananas  did  each  boy 
get? 

5.  A  farmer  had  60  sheep.  How  many  did  he 
have  after  selling  20  sheep? 

6.  It  is  2  miles  from  Arthur's  home  to  his  aunt's. 
On  Saturday  Arthur  made  two  trips  on  his  bicycle  to 
his  aunt's  and  return.     How  many  miles  did  he  ride  ? 

7.  If  it  takes  3  yards  of  cloth  to  make  one  apron, 
how  many  aprons  can  be  made  from  12  yards  ? 

a  If  oranges  sell  at  20^  a  dozen,  how  many  dozen 
can  be  bought  for  60^  ? 

9.  What  is  the  cost  of  3  pounds  of  coffee  at  30^  a 
pound? 

10.  What  is  the  cost  of  five  2-cent  stamps? 

11.  At  60  cents  a  yard,  how  much  will  |  yd.  of 
cloth  cost? 

12.  How  many  gallons  are  'there  in  12  quarts? 


WRITTEN  PROBLEMS  111 

145.    Written  Exercises. 


a 

b 

C 

d 

e 

/ 

(1 

h 

474 

938 

546 

218 

917 

645 

539 

462 

907 

794 

477 

827 

129 

387 

958 

547 

836 

672 

259 

388 

451 

719 

867 

243 

593 

866 

998 

916 

679 

693 

926 

589 

688 

978 

738 

193 

832 

568 

775 

362 

966 

489 

999 

778 

178 

686 

944 

318 

389 

749 

823 

143 

819 

617 

233 

759 

146.   Written  Problems. 

1.  Harry  is  saving  his  money  to  buy  a  bicycle  that 
will  cost  $  45.  He  has  saved  $  38.  How  much  more 
must  he  save  before  he  can  pay  for  the  bicycle  ? 

2.  Four  boys  went  fishing.  They  paid  25^  for  the 
use  of  a  boat,  15^  for  bait,  35^  for  some  lines,  and 
45^  for  lunch.  Find  the  whole  cost  of  the  trip.  Find 
each  boy's  share  of  the  expenses. 

3.  A  man  bought  a  horse  for  $95.  For  what  must 
he  sell  the  horse  to  gain  $25  ? 

jiv.  A  boy  had  60  marbles  and  sold  one  third  of 
them.  How  many  marbles  did  he  sell  ?  How  many 
had  he  left  ? 

5.  A  farmer  sold  three  cows  for  the  following  sums : 
$  28,  $  36,  and  $  40.    How  much  did  he  get  for  them  ? 

6.  A  farmer  sold  4  cows  at  $32  each.  How  much 
did  he  get  for  them  ? 


il2  ADDITIOxV   AND   SUBTRACTION 

SURFACES 


ABC  D 

147.   1.    The  surface  of  Fig.  A  is  —  of  the  surface 
of  Fig.  B. 

2.  The  surface  of  Fig.  ^  is  —  of  the  surface  of 
Fig.  C. 

3.  The  surface  of  Fig.  B  is  —  of  the  surface  of 

Fig.  a 

4.  The  surface  of  Fig.  ^  is  —  of  the  surface  of 
Fig.  D. 

5.  The  surface  of  Fig.  ^  is  —  of  the  surface  of 
Fig.  D, 

6.  The  surface  of  Fig.  ^  is  —  times  the  surface 
of  Fig.  A. 

7.  The  surface  of  Fig.  D  is  four  times  the  sur- 
face of  Fig.  — . 

8.  The  surface  of  Fig.  D  is  two  times  the  surface 
of  Fig.  — . 

9.  The  ratio  of  Fig.  A  to  Fig.  5  is  — . 

10.  The  ratio  of  Fig.  C  to  Fig.  A  is  — . 

11.  What  part  of  the  surface  of  Fig.  C  is  equal  to 
the  surface  of  Fig.  A  ? 

12.  Three  times  Fig.  A  is  equal  to  Fig.  — . 

13.  If  Fig.  A  represents  2  square  inches,  Fig.  B 
will  represent  —  square  inches. 


LENGTH  113 

148.   1.    How  long  is  this  book  ? 

2.  The  unit  of  length  used  to  measure  short  dis- 
tances is  the  inch. 

3.  How  long  is  this  room  ? 

The  foot  is  the  imit  of  measure  next  in  length  to 
the  inch.  We  use  the  unit  1  foot  in  measuring  the 
length  of  a  room. 

4.  What  is  the  unit  of  length  in  measuring  cloth  ? 

5.  The  rod  and  the  mile  are  each  units  of  length. 
These  are  used  in  measuring  long  distances. 

6.  Study  the  inch,  the  foot,  and  the  yard.     See 
Lesson  VII,  p.  21. 

7.  Draw  a  line  12  inches  long.     Divide  the  line 
into  inches.     The  unit  of  measure  of  the  line  is  — . 

8.  Draw  a  line  3  feet  long. 

9.  Three  feet  are  1  yard. 

10.  One  foot  is  —  third  of  a  yard. 

11.  In  5  feet  there  is  —  yard  and  —  feet. 

12.  What  part  of  one  foot  is  one  inch  ? 

13.  Six  inches  are  —  twelfths  of  12  inches. 

14.  Divide  12  inches  into  3  inches. 

15.  In  12  inches  there  are  —  3  inches. 

16.  Divide  12  inches  into  4  inches. 

17.  In  12  inches  there  are  —  4  inches. 

18.  Memorize : 

Twelve  inches  are  one  foot, 
Three  feet  are  one  yard. 

1st  P.k  Ahitii— S  ^ 


114  ADDITION  AND  SUBTRACTION 

149.   1.   Your  desk  top  has  length.     Has  it  width  ? 

2.  Anything  that  has  length  and  width  has  area. 

3.  Have  the  sides  of  this  room  area  ? 

4.  Has  a  book  cover  area  ? 

5.  A  square  inch  is  a  square  whose  sides  are  each 
one  inch.     Draw  a  square  inch. 

6.  A  square  inch  is  the  smallest  unit  of  area. 

7.  Name  a  unit  of  area  larger  than  the  square  inch. 

a  Draw  an  oblong  3  in.  long  and  1  in.  wide. 
Divide  the  oblong  into  square  inches.  How  many 
square  inches  are  there  in  the  oblong? 

9.  Draw  an  oblong  3  in.  long  and  2  in.  wade. 
Divide  it  into  square  inches.  How  many  square 
inches  are  there  in  the  oblong  ?  What  is  the  area  of 
the  oblong?  How  many  3  sq.  in.  are  there  in  the 
oblong  ? 

10.  In  Problem  9,  two  square  inches  are  what  part 
of  the  oblong  ?  Three  square  inches  are  what  part  of 
the  oblong  ? 

11.  Draw  an  oblong  4  in.  long  and  3  in.  wide.  How 
many  4  sq.  in.  can  be  made  of  the  oblong  ?  What 
is  the  area  of  the  oblong  ? 

12.  Draw  an  oblong  4  in.  long  and  2  in.  wide.  How 
many  4  sq.  in.  can  be  made  of  this  oblong?  What 
is  the  area  of  the  oblong  ? 

13.  Draw  an  oblong  4  in.  long  and  wide  enough  to 
contain  4  sq.  in.   What  is  its  area  ? 


CHAPTER  IV 

MULTIPLICATION  AND   DIVISION 

SIMPLE  FRACTIONS,  COMPOUND  NUMBERS,   REVIEWS 

MULTIPLICATION  —  LESSON  0 

150.   Oral  Problems.* 

1.  How  much  will  2  chairs  cost  at  $  4  each  ? 
Model  for  oral  recitation :  Since  1  chair  costs  $  4, 

2  chairs  will  cost  2  times  $  4,  or  $  8. 
Model  for  written  recitation  : 

$  4,  cost  of  1  chair. 

x2 

$  8,  cost  of  2  chairs. 

2.  How  much  will  2  clocks  cost  at  $  6  each  ? 

3.  At  $  3  a  pair,  how  much  will  2  pairs  of  shoes 
cost? 

4.  How  much  will  2  tables  cost  at  $  5  each  ? 

5.  At  6^  each,  how  much  will  2  oranges  cost  ? 

6.  How  much  will  3  hats  cost  at  $  4  each  ? 

7.  There  are  4  quarts  in  a  gallon.  How  many 
quarts  are  there  in  2  gallons  ? 

*  Drill  should  be  given  upon  these  and  similar  problems  until  the  pupils 
are  familiar  with  the  language  forms  used  in  the  analysis.  The  written 
form  should  be  taken  up  after  the  oral  form  has  been  mastered.  Apply 
these  forms  to  similar  problems  on  the  succeeding  pages  of  the  text. 

115 


116  MULTIPLICATION   AND   DIVISION 

151.   Oral  Problems.* 

1.  If  2  chairs  cost  $  6,  what  is  the  cost  of  1  chair  ? 

Model  for  oral  recitation:    If  2  chairs  cost  $6, 
1  chair  will  cost  one  half  of  $6,  or  $3. 
Model  for  written  recitation : 

$  3,  cost  of  1  chair. 
2)$G,  cost  of  2  chairs. 

2.  If  2  stoves  cost  $  10,  what  is  the  cost  of  1  stove  ? 

3.  If  2  tables  cost  $  8,  what  is  the  cost  of  1  table  ? 

4.  If  2  tablets  cost  12^,  what  is  the  cost  of  1  tablet  ? 

5.  If  3  boxes  of  berries  cost  15^,  what  is  the  cost 
of  1  box  ? 

6.  If  3  pencils  cost  6^,  what  is  the  cost  of  1  pencil  ? 

7.  If  3  hats  cost  $  9,  what  is  the  cost  of  1  hat  ? 

8.  If  4  pairs  of  shoes  cost  $12,  what  is  the  cost 
of  1  pair? 

9.  If  1  yd.  of  cloth  costs  12^,  what  is  the  cost  of 

iyd.?  . 

10.  If  1  yd.  of  ribbon  costs  8^,  what  is  the  cost  of 
iyard? 

11.  What  is  the  cost  of  i  yd.  of  cloth  at  15^  a  yard  ? 

12.  What  is  the  cost  of  h  doz.  eggs  at  12^  a  dozen  ? 

13.  If  4  chairs  cost  $  12,  what  is  the  cost  of  1  chair  ? 

14.  What  is  the  cost  of  1  stove  at  the  rate  of  2 
stoves  for  $12? 

*  See  note,  p.  115. 


MULTIPLICATION  — LESSON   C  117 

152.   Oral  Exercises. 

1.  Add  a  coliunn  of  four  4's  ;  of  five  3's. 

2.  Count  by  4's  to  16  ;  by  3's  to  15. 

3.  Add  a  column  of  seven  2's  ;  of  eight  2's. 

4.  Count  by  2's  to  14  ;  by  2's  to  16. 

5.  Four  4's  are  — ;  five  3's  are  — ;  three  5's  are — . 

6.  Seven  2's  are  — ;  two  7's  are  — ;  eight  2's  are  — . 

7.  In  16  there  are  —  4's.    There  are  —  2's  in  16. 

8.  How  many  nickels  are  there  in  15  cents? 

9.  Memorize : 

4  8  7  2  2 

x4  x2  x2  x8  x7 

Te  16  14  16  14 

10.  7  is  read  two  7's  are  14.     It  may  also  be 
X  2     read,  two  times  7  is  14. 

11.  A  boy  bought  3  oranges  at  5  )^  each.    How  much 
did  he  pay  for  all  ? 

12.  At  3  ^  each,  how  much  will  5  pencils  cost  ? 

13.  Frank  has  $4  and  Arthur  has  four  times  as 
much  money.     How  much  money  has  Arthur? 

14.  At  1 2  each,  how  much  will  8  hats  cost? 

15.  There  are  7  days  in  one  week.     How  many 
days  are  there  in  2  weeks? 

16.  Etiiel  worked   8   problems  and   Edna  worked 
twice  as  many.     How  many  problems  did  Edna  work  ? 


118 


MULTIPLICATION  AND  DIVISION 


153.  Study  Exercises. 

4                 8                 7 

2 

2 

x4              x2              x2 

,x8 

x7 

Study  the  above  exercises  until  you  can  give  the 
products  without  hesitation. 

Give  the  products  from  right  to  left,  adding  3  to 
each  product,  thus:  14,  17;  16,  19 ;  etc. 


154.  Written  Exercises. 

abed 

e 

•     / 

9 

1.  2034     3140     4213     1324 

4321 

8400 

2341 

x4        x4        x4        x4 

x4 

x4 

x4 

2.  3023 

3333 

2323 

2030 

1302 

3032 

3021 

x5 

x5 

x5 

x5 

x5 

x5 

x5 

3.  1212 

2021 

2222 

2012 

1220 

2222 

2121 

x6 

x6 

x7 

x7 

x7 

x8 

x8 

Multiply  457  by  20. 


Model 

:     457 

x20 

9140 

a 

b 

3457 

5678 

x20 

x20 

0  times  457  is  0.    Write  0  under 
the  0  in  the  multiplier.     2  tunes 
7  is  14.     Write  the  4  under  the  2 
in  the  multiplier,  and  continue. 
c  d  e  f  q 

6785     3467     4576     6587     3478 
x20      x20      x20      x20      x  20 


6.  3425 
x30 


5243  4035  2304  4230  3040  2130 
x30   x30   x40   x40   x  40   x  50 


ORAL  EXERCISES  119 

155.    Oral  Exercises. 

Add  each  column  as  indicated  in  Step  C,  p.  49. 


a 

b 

c 

d 

e 

/ 

9 

A 

^ 

J 

k 

I 

TO 

7 

8 

8 

7 

9 

8 

7 

6 

9 

8 

7 

4 

2 

3 

7 

2 

3 

1 

2 

3 

4 

1 

2 

3 

6 

8 

6 

5 

8 

8 

4 

4 

5 

5 

3 

8 

7 

4 

2 

4 

8 

1 

2 

6 

6 

5 

5 

7 

2 

5 

3 

8 

7 

7 

8 

7 

9 

8 

7 

6 

9 

6 

5 

7 

2 

3 

5 

2 

3 

1 

2 

3 

4 

1 

4 

3 

6 

8 

6 

5 

8 

8 

4 

8 

5 

5 

3 

6 

5 

3 

8 

7 

8 

9 

7 

9 

4 

.7 

6 

9 

8 

6 

5 

4 

156.     1.   Show  by  objects  how  many  2  books  there 
are  in  3  books ;  in  5  books ;  in  7  books. 

2.  Show  how  many  3  boys  there  are  in  7  boys. 

3.  In  7  there  are  —  2's  and  —  remainder. 

4.  How  many  2's  are  there  in  70  ? 

Model  :      35         In  7  there  are  three  2's  and  1 

2)70.    remainder.     Write  3  above  the  7, 

and  think  the  1  before  0.     In  10 

there  are  5  twos.     Write  5  above  the  0.     There  are 

35  twos  in  70.     One  half  of  70  is  — . 


157.    Divide: 

a               b              c              d               e 

/ 

1-    2)70       2)50       2)92       2)76       2)34 

3)42 

2.    3)102     3)105     3)300     3)420     3)720 

3)672. 

3.    2)530     2)302     2)710     2)930     3)1032     3)7032. 


120  MULTII'LICATION    AND   DIVISION 

DIVISION  — LESSON   C 

158.  .1.    In  $4  there  are  —  $2.     In  $5  there  are 
—  $2  and  $  —  reniainder. 

2.  What  is  1  of  5  ?     What  is  l  of  11  ? 

3.  To  find  one  half  of  a  number,  divide  it  by  — . 

4.  Find  i  of  9. 

Model:        4^        There   are   four  2's   in    9,  and 

2)9.     1   remainder.     The    remainder   is 

written  over  the  divisor  as  above. 

5.  One  half  of  9  is  — .     Nine  divided  by  2  is  — . 

6.  Find  1  of  7  ;  of  11 ;  of  13 ;  of  5  ;  of  10. 

7.  Find  1  of  4 ;  of  7  ;  of  10 ;  of  5  ;  of  8. 

a    Find  J-  of  5 ;  of  6 ;  of  7  ;  of  8  ;  of  9 ;  of  10. 
9.    Six  divided  by  2  may  be  written :  2)6,  or  6  -i-  2, 
orf. 

159.  Bead  and  memorize : 


4                2 
4)16           8)16 

2 

7)14 

7 

2)14 

8 
2)16 

160.    Supply  quotients  in  the  following : 

1.    6^2  =  .r. 

7.    10^2  =  a:. 

13. 

15^-3  =  0;. 

2.     S-^i=X. 

a   12^6  =  a:. 

14. 

12^2  =  x. 

3.    6^3  =  a:. 

9.   15^5  =  x. 

15. 

U-i-2  =  x. 

4.    9-5-3  =  a;. 

10.    16-5-2  =  a:. 

16. 

16^8  =  x'. 

5.    8-^2-a:. 

11.    12^3  =  a:. 

17. 

14  -*-  7  =  X. 

6.    4-^2  =  0:. 

12.    16H-4=ic. 

18. 

12-h4  =  x. 

161.    Drill  Exercises. 


1. 


a 


2.     ^ 


6  12 


3  3 


3.     ^ 
4 


0  1_5 

'5 


5. 


0  JL^ 

3  4 


-12. 
6 


H.  1_2_ 

4  2 


\TIOX- 

-LESSON    D 

121 

Give 

answers  : 

C 

d 

e 

/ 

14 

13 

4 

3 

-2" 

-3- 

3 

2 

15 

14 

5 

5 

3 

3 

3 

2 

16 

10 

7 

5 

"4" 

"3 

3 

4 

16 

11 

8 

6 

-2' 

-3" 

3 

4 

16 

11 

7 

7 

~8- 

2" 

2 

4 

14 

13 

9 

9 

7 

2 

2 

4 

162.  Subtract: 

a             6             c  d             e             f  9 

1.  8457  9745  9944  '4715  6486  9312  7458 
2958  3887  3146  _859  2688  6317  2769 

2.  9236  5211  8452  8294  9732  8290  7408 
8267  2343  _674  2705  5846  1304   980 

MULTIPLICATION  —  LESSON   D 

163.  1.    Add  a  column  of  six  3's ;  of  nine  2's. 

2.  Two  9's  are  — ;  nine  2's  are  — . 

3.  Find  the  sum  of  three  7's ;  of  seven  3's. 

4.  In  3  weeks  there  are  —  days. 

164.  Memorize: 

6  9  7  3  2        •    3 

x^  x^x3  x6  x9  x7 

18         Ts  li  18  18  2T 


122  MULTIPLICATION   AND  DIVISION 


165.   Study  Exercises. 

6            9            7 

3 

2 

3 

x3         x2         x3 

x6 

x9 

x7 

Study  the  above  exercises  until  you  can  give  the 
products  without  hesitation. 

Give  the  products  in  the  above  exercises  from  right 
to  left. 

Give  the  products  from  right  to  left,  adding  3, 
4,  and  5  to  each  product,  thus,  adding  3  :  21,  24 ; 
18,  21;  etc. 

Give  the  products  from  right  to  left,  adding  6, 
7,  and  8  to  each  product,  thus,  adding  6:  21,  27; 
18,  24;  etc. 


166.   Written  Exercises. 

iviuitipiy : 

a               b 
1.    7897      9789 
2            2 

c 
8789 
2 

6798 
2 

• 

e 

7968 
2 

/ 

9687 
2 

2. 

5467 
3 

7654 
3 

4567 
3 

7456 
3 

6745 
3 

5764 
3 

3. 

4321 
4 

1423 
4 

3012 
5 

2130 
5 

2301 
6 

3210 
6 

4. 

3023 

.7 

2220 
8 

2323 

7 

3020 
6 

2343 
4 

3223 
5 

5. 

7605 
20 

6750 
30 

4032 
40 

3120 
60 

3213 
50 

1203 
50 

DIVISION  — LESSON  D  128 


DIVISION —  LESSON   D 


167.    1.    Memorize  tlie  following  : 

6    •       _9  _7  _3  _2  _3 

3p         2)T8         3)21  6)18         9)18  7)2l 

2.  A  boy  had  18  marbles  and  sold  \  of  them. 
How  many  marbles  did  he  sell  ?  How  many  marbles 
did  he  have  left  ? 

3.  How  many  weeks  are  there  in  21  days  ? 

4.  A  girl  had  18  cents.  She  spent  \  of  her  money 
for  some  paper.  What  was  the  cost  of  the  paper  ? 
How  much  money  had  she  left  ? 

5.  If  18  apples  are  divided  equally  among  six  boys, 
what  part  of  the  whole  number  of  apples  will  each 
boy  receive  ? 

6.  If  18  pencils  are  divided  into  2  equal  groups, 
Jiow  many  pencils  will  there  be  in  each  group  ? 

7.  What  part  of  18  inches  are  6  inches  ? 

a   What  is  the  ratio  of  21  to  7  ?     Of  7  to  21  ? 

9.  In  finding  the  number  of  $3  there  are  in  $21, 
the  unit  of  measure  is  — ,  and  the  quantity  measured 
is  — . 

10.  What  part  of  18  feet  are  2  feet  ?     Are  9  feet  ? 

11.  What  is  the  ratio  of  2  feet  to  18  feet  ? 

12.  Find!  of  $21.     Find  1  of  $18. 

13.  How  many  2-cent  stamps  can  be  bought  for 
18  cents  ? 


124 


MULTIPLTCATIOX   AND   DIVISION 


168.  Study  Exercises. 

3)l8        2)"T8        3)2l  6)18"  9)18'  7)2T 

3)T9        2)19        3)22  6)19  9)l9  7)22 

3)"20        2)17       3)"23  6)l3  5)l6  7)23 

3)16        3)17        8)17  4)17  7)15  7)l6 

In  the  study  of  the  above  exercises  use  the  follow- 
ing models : 

(a)  In  19  there  are  six  3's  and  1  remainder. 
(h)  One  third  of  19  is  6J. 

Study  the  above  exercises  until  you  can  give  the 
quotients  without  hesitation. 

169.  Written  Exercises. 


a 

6 

c 

d 

e 

1. 

2)1980 

2)1330 

2)9398 

2)5112 

2)1776 

2. 

2)1816 

2)1412 

2)1306 

2)3170 

2)1360' 

3. 

3)1005 

3)1215 

3)1998 

3)1665 

3)2322 

4. 

3)2001 ' 

3)1710 

3)1671 

3)1356 

3)5109 

S. 

4)1768 

4)1372 

4)1736 

4)1216 

4)9736 

6. 

5)1160 

5)1615 

5)1510 

5)1155 

5)1665 

7. 

6)1812 

6)1998 

6)1332 

6)7206 

6)7938 

8. 

7)2331 

7)1631 

7)9184 

7)1561 

7)7140 

9. 

8)1776 

8)9768 

8)1608 

8)1760 

8)1696 

10.  9)1809   9)1998   9)1098   9)1089   9)1908 


DIVISIOX  — LESSON   D  125 

170.    Oral  Problems.* 

1.  At  $2  each,  how  many  chairs  can  be  bought 
for  $8? 

Model  for  oral  recitation :  Since  1  chair  costs  $  2, 
as  many  chairs  can  be  bought  for  $8  as  there  are  $2 
in  $8,  or  4.     Four  chairs  can  be  bought  for  $8. 
Model  for  written  recitation  : 

_4  chairs  for  $8. 
cost  of  1  chair,  $2)$  8 

2.  If  1  pair  of  shoes  costs  $  3,  how  many  pairs  can 
be  bought  for  $12? 

3.  If  1  box  of  berries  costs  5,^,  how  many  boxes 
can  be  bought  for  20^? 

4.  At  $3  each,  how  many  hats  can  be  bought  for 
$9? 

5.  How  many  yards  of  ribbon  at  4^  a  yard  can  be 
bought  for  16^? 

6.  At  6^  each,  how  many  tablets  can  be  bought 
for  18^? 

7.  If  a  boy  earns  $  4  a  week,  in  how  many  weeks 
will  he  earn  $20? 

8.  A  farmer  sold  some  sheep  for  $6  each.     He 
received   $18.     How  many  sheep  did  he  sell? 

9.  If  2  girls  sit  in  each  seat,  how  many  seats  will 
20  girls  occupy  ? 

10.    A  girl  earns  8^  a  day.    In  how  many  days  will 
she  earn  16^? 

*  See  note,  p.  115. 


126  MULTIPLICATION   AND   DIVISION 

171.    Written  Problems. 

1.  A  man  bought  4  cows.  For  one  he  paid  $27, 
for  another  $32,  for  another  $36,  for  another  $40. 
How  much  did  he  pay  for  the  four  cows  ? 

2.  At  $32  each,  what  will  be  the  cost  of  4  cows? 

3.  Can  you  find  the  cost  of  the  cows  in  the  first 
problem  by  multiplication  ? 

4.  Can  you  find  the  cost  of  the  cows  in  the  second 
problem  by  addition  ? 

5.  When  can  you  use  either  addition  or  multipli- 
cation to  find  the  cost  of  cows  ? 

6.  When  you  can  use  either  addition  or  multipli- 
cation, which  is  the  better  to  use  ?     Why  ? 

7.  A  man  bought  a  farm  for  $3675,  and  sold  it  for 
$  5000.    Did  he  gain  or  lose,  and  how  many  dollars  ? 

8.  A  man  bought  6  sheep  at  $3  each  and  sold 
them  for  $  25.     Did  he  gain  or  lose,  and  how  much  ? 

9.  A  farmer  owned  1 860  acres  of  land.  He  divided 
his  land  into  3  farms,  with  the  same  number  of  acres 
in  each.  How  many  acres  were  there  in  each  of  the 
farms  ? 

10.  At  $4  each,  how  many  sheep  can  be  bought 
for  $128? 

11.  At  $  5  each,  how  many  barrels  of  flour  can  be 
bought  for  $65? 

12.  At  23^  a  yard,  what  will  be  the  cost  of  6  yards 
of  cloth  ? 


ORAL   PROBLEMS  127 

172.    Oral  Problems. 

1.  How  many  halves  are  there  in  1  apple  ?  In  1 
circle  ?   In  1  dollar  ?   In  1  day  ? 

2.  How  many  halves  are  there  in  2  apples  ?  In  2i 
apples  ?   In  3  apples  ?   In  o^  apples  ?    In  4  J  days  ? 

3.  How  many  apples  must  be  cut  into  halves  in 
order  to  get  |  apples  ?   |  apples  ?    |  apples  ? 

4.  Tell  how  many  whole  apples  each  of  the  fol- 
lowing is  equal  to :  |-  apples,  |-  apples,  |  apples. 

5.  A  man  paid  4  boys  |  dollar  each  for  helping 
him  on  a  Saturday.  How  many  dollars  did  he  pay 
them  all  ? 

6.  There  were  8  children  at  a  party.  A  lady  gave 
one  half  of  an  orange  to  each.  How  many  oranges 
did  she  give  to  all  ? 

7.  2^  apples  =  f  apples ;  |-  apples  =  —  J  apples ; 
3^  apples  =  f  apples. 

8.  4i  days  =  |  days ;  5 J  dollars  =  |  dollars. 

9.  6 1  feet  =  I  feet ;  1 J  inches  =  f  inches. 

10.  f  apples  =  —  apples ;  ^  oranges  =  —  i  oranges. 

11.  Tell  how  many  half  apples  each  of  the  follow- 
ing is  equal  to :  2  apples,  2^  apples,  4  apples,  41 
apples,  5^  apples,  31  apples,  6  apples. 

12.  Seven  half  dollars  are  equal  to  how  many 
dollars  ? 

13.  A  girl  spent  5  half  days  in  the  city.  How  much 
more  than  2  days  did  she  spend  there  ? 


128  MULTIPLICATION   AND   DIVISION 

173.  Drill  Exercises. 

G  X  2  is  read,  6  multiplied  hy  2.   It  is  the  same  as     t/ 

Supply  products  for  x,  and  add  to  each  product  the 
number  above  the  column  as  in  a :  12,  14  ;  IG,  18. 
Give  answers  only : 

(2)  (3)  ■       (5)  (7) 

a  b  erf 

1.  Gx2  =  12,14  8x2  =  16,19  7x3  =  a:  3x2  =  a: 

2.  4x4  =  lG,18  4x3  =  x  2x5  =  x     oxS  =  x 

3.  5x2  =  x  6x3  =  a:  4x2  =  x  3x4  =  a; 

4.  3x3  =  a;  7x2  =  x  3x5  =  a:  2x6  =  x 

5.  2x3  =  a;  Sx(j  =  x  9x2-a;  2x8  =  a; 

174.  Drill  Exercises. 

Supply  quotients  in  place  of  x. 


a 
1.    18^3  =  a: 

b 
12H-4  =  a; 

c 
16^2  =  aj 

d 
19-^3  =  a: 

2.    14-^7  =  0: 

lS-^G  =  x 

18-4-9  =  a: 

11^2  =  a; 

3.    21^3  =  aj 

16^8  =  ^ 

15-^3  =  x 

13H-6  =  a; 

175.   Written  Exercises. 

a                    b                     c 

1.    978,978     897,897     679, 
x2             x2 

d 

769     587,857 
x2             x2 

e 
459,596 
x2 

2.  765,765  657,657  547,574  637,673  564,767 
x3      x3  x3     x3     x3 

3.  324,243  432,432  304,203  321,321  123,123 

x4  x4             x4  x5  x6 


ORAL   EXERCISES  129 

176.    Oral  Exercises. 

Add  each  column  as  indicated  in  Step  C,  p.  49. 


a 

b 

c 

d 

e 

/ 

9 

h 

I 

i 

/t 

I 

m 

3 

9 

8 

7 

6 

G 

1 

7 

6 

9 

9 

5 

7 

5 

3 

2 

4 

1 

3 

9 

3 

4 

1 

3 

5 

6 

5 

7 

8 

6 

9 

t 

8 

7 

1 

4 

6 

1 

9 

7 

5 

9 

3 

8 

4 

2 

3 

9 

6 

4 

4 

3 

5 

3 

2 

A 

1 

5 

8 

6 

8 

7 

8 

6 

7 

5 

7 

8 

6 

9 

5 

4 

4 

2 

3 

2 

8 

3 

4 

8 

9 

7 

8 

9 

9 

8 

2 

2 

7 

9 

7 

8 

6 

7 

8 

9 

6 

6 

8 

9 

8 

6 

9 

8 

Name  five  combinations  whose  smns  are  ten. 

When  these  combinations  occur  in  a  column,  they 
may  be  taken  together.  Exercise  a  above  may  be 
added :  12,  22,  29,  39,  42.  Add  the  above  exercises 
in  a  similar  manner. 


177.    Written  Exercises. 

Sul^tract : 
a                     h 
16,043         72,345 
9,876         34,567 

c 
90,234 

27,840 

d 
45,803 
14,769 

e 
84,087 
28,309 

178.    Oral  Exercises. 

Add  a  column  of  ten  3's ;  of  five  4's ;  of  five  5's. 

Memorize : 

8            9  5  3  3  4  5 

x3          x3  x4  x8  x9  x5  x5 

~24         17  "20  "24  "27  "20  "25 

IsT  liK  Anrni — 0 


MULTIPLICATION   AND   DIVISION 


179. 

8 
x3 


MULTIPLICATION  —  LESSON  E 
Study  Exercises. 


9 
x3 


x4 


3 

x8 


3 

x9 


4 

x  5 


X  o 


Study  the  above  exercises  until  you  can  give  the 
products  without  hesitation. 

Give  the  products,  from  right  to  left ;  adding  4  to 
each  product ;  adding  6  to  each  product. 


M^  180 


180.    Multiply  3457  by  23. 


DEL  :     3457         Multiply  3457  first  by  3,  and 

23     write  the  product.    Next,  multiply 

10371     by  2.    Two  7's  are  14.    Write  the 

6914       4  under  the  2.    After  completing 

79511     the  multiplication  by  2,  draw  a 

line  and  add  the  products. 


181 


Written  Exercisei 
Multiply : 


1.  3457x23 

2.  6789x23 

3.  9876x32 

4.  5647x32 
5,^8975x23 

6.  3240x35 

7.  6978x32 

8.  4036x23 

9.  9380x32 


10.  3243x45 

11.  4545  X  54 

12.  3454x45 

13.  2050x54 

14.  3524x54 

15.  2320x67 

16.  3023x67 

17.  1323x67 

18.  2032x67 


19.  3223x89 

20.  2130x98 

21.  3231x89 

22.  3020x98 

23.  3123x84 

24.  2323x29 

25.  4534x15 

26.  3750x13 

27.  4567x20 


ORAL   riiOBLEMS  131 

182.    Oral  Problems. 

1.  At  $3  each,  how  many  chairs  can  be  bought 
for  $18? 

2.  How  much  will  4  tables  cost  at  $  5  each  ? 

3.  If  8  hats  cost  $16,  how  much  w411  1  hat  cost? 

4.  At  $  4  each,  how  many  desks  can  be  bought 
for  $12? 

5.  How  much  will  6  tablets  cost  at  3  ^  each  ? 

6.  How  many  pencils  at  2^  each  can  be  bought 
for  8^?  :h 

7.  At  the  rate  of  4  for  12^,  what  is  the  cost  of  1 
pencil  ? 

8.  If  3  packages  of  popcorn  cost  15y,  what  is  the 
cost  of  1  package  ? 

9.  How  much  will  3  boxes  of  berries  cost  at  6y  per 
box? 

10.  Four  boys  rode  on  a  street  car.  They  paid  5^ 
each.     How  much  did  it  cost  the  four  boys  ? 

11.  Mary  spent  3  weeks  with  her  aunt.  How 
many  days  did  she  spend  with  her  ? 

12.  George  paid  21^  for  3  tablets.  How  much  was 
this  for  each  tablet  ? 

13.  Ethel  walks  2  miles  to  school.  How  far  does 
she  walk  in  going  and  coming  each  day  ?  How  far 
does  she  walk  in  1  week  ? 

14.  Frank  is  7  years  old.  His  brother  is  21  years 
old.     His  brother  is  how  many  times  as  old  as  Frank  ? 


132  MIILTIPLICATIOX    AND   DIVISION 

183.    Written  Problems. 

1.  A  boy  I'ode  15  miles  in  3  hours.  What  was  the 
average  rate  per  hour  ? 

2.  A  train  runs  96  miles  in  4  hours.  What  is  the 
average  rate  per  hour  ? 

3.  A  grocer  sold  $  1380  worth  of  goods  in  6  days. 
Find  the  average  daily  sales  for  the  week. 

4.  George  weighs  84  lb. ;  Walter  weighs  76  lb. 
How  much  will  the  two  boys  together  weigh  ?  What 
is  the  average  weight  of  the  two  boys  ?  Which  of  the 
boys  weighs  more  than  the  average  weight  ? 

5.  There  are  50  pupils  in  Room  A ;  43  pupils 
in  Room  B;  40  pupils  in  Room  C;  and  39  pupils 
in  Room  D.  How  many  pupils  are  there  in  the 
four  rooms  ?  Find  the  average  number  of  pupils  in 
a  room.  Which  of  the  rooms  have  more  than  the 
average  number  of  pupils?  Which  have  less  than 
the  average  number  ?  Has  any  room  the  average 
number  of  pupils  ? 

6.  A  farmer  sold  a  grocer  7  lb.  of  butter  at  23  ^  a 
pound.  What  was  the  value  of  the  butter?  The 
farmer  bought  of  the  grocer  2  lb.  of  coffee  at  25  ^  a 
pound,  and  3  lb.  of  tea  at  35^  a  pound.  What  was 
the  value  of  the  coffee  and  tea  ?  Did  the  farmer  owe 
the  grocer  or  the  grocer  owe  the  farmer,  and  how 
much  ? 

7.  Find  i  of  $1760.     Find  ^  of  $1600. 

8.  Which  is  the  more,  i  of  $  120  or  |  of  $  176  ? 


DIVISION  — LESSON   E  133 


DIVISION  — LESSON  E 

184.    1.    Memorize  tlte  following  : 

8           9           5           3           3 

4           5 

3)24      3)27      4)20      8)24      9)27 

5)20      5)25 

2.    Eight  is  ^  of  — .     5  is  i  of  — . 

3  is  1  of 

3.    Three  is  l  of  — .     4  is  i  of  — . 

5  is  1^  of  — 

4.  What  is  i  of  $  24  ?     What  is  i  of  $  25  ? 

5.  What  is  i  of  20?     What  is  1  of  27? 

6.  In  27  there  are  —  3's.     In  28  there  are  —  3's, 
and  —  remainder. 

7.  In  24  there  are  —  3's.     In  25  there  are  —  3's, 
and  —  remainder. 

8.  What  is  1  of  20?     What  is  1  of  21? 

9.  What  is  tlie  ratio  of  5  to  20  ?     Of  20  to  5  ? 

10.  How  many  $  5  units  are  there  in  $  25  ? 

11.  How  many  $1  imits  are  there  in  $25? 

12.  How  many  units  of  3  feet  are  there  in  24  feet  ? 

13.  How  many  times  must  the  yardstick  be  applied 
in  measuring  24  feet? 

14.  Nine  feet  are  i  of  — .     $4  are  ^  of  $ — . 

15.  There  are  24  hours  in  one  day.  How  many 
hours  are  there  in  l  of  a  day  ? 

16.  A  boy  had  20^.  He  bought  a  tablet  that  cost  \ 
of  his  money.  W^hat  was  the  cost  of  the  tablet  ? 
The  boy  had  — ^  left. 

17.  What  is  1  of  20^  ?     What  is  1  of  20^  ? 


134  MULTIPLICATION  AND  DIVISION 

185.  Study  Exercises. 

3)24      3)27      4)20    .  8)"24  9)27  5)"20  5)"25 

3)25      3)"28      4)"21      8)25  9)18  5)'2T  5)26 

3)26      3)29      4)22      8)26  9)29  5)"22  5)27 

Study  the  above  exercises  until  you  can  give  the 
quotients  without  hesitation. 

Give  the  quotients  and  the  remainders,  if  any. 

Give  the  quotients  with  the  remainders  expressed 
as  fractions.     Review  Exercise  168,  p.  124. 

186.  Written  Exercises. 

Use  the  numbers  above  the  columns  as  divisors : 


q  9 

4,  3 

5 

6 

1. 

2^2,724. 

7. 

201,620 

13. 

202,520 

19. 

181,213 

2. 

272,427 

8. 

177,808 

14. 

252,025 

20. 

139,392 

3. 

251,515 

9. 

217,371 

15. 

152,520 

21. 

738,798 

4. 

266,664 

10. 

222,200 

16. 

101,520 

22. 

193,278 

5. 

267,267 

11. 

982,208 

17. 

252,015 

23. 

792,192 

6. 

287,878 

12. 

140,142 

18. 

267,676 

24. 

180,192 

187.    Write  in  a  column  and  add  : 

1.  $1045,  $72.05,  $108.75,  $9.18,  $.75,  $704. 

2.  $304.50,  $40.20,  $1000,  $.85,  $19.90,  $1.25. 

3.  $6.40,    $200.45,   $3.05,    $89.26,    $6,   $600, 
$8.30. 

4.  $300,  $8,  $4000,  $.12,  $20,  $10.50,  $.64. 


DRILL   EXERCISES  135 

188.  Drill  Exercises. 

Give    quotients    and    express    the    remainders    as 
fractions : 

1       _2Jl     2_1     _2_2  c       2_0     _2_1     _2_2  -.^        2_7_     2_8      2  9 

J-       4  ?     4  ?     4--  o-        5   9  "5  ?     5   •  •'"*••        9  ?     9  ?     9  • 

o       1_5_     1_6     1_1  „       1_6      1_7_     18.  TO       10      17      1_8 

^'       35353*  '•       45454-  ■'■^-        8"?     8"5  "8   • 

o       2_5      2_6      2_7  a      JL8_     1_9.     2_0_  to       2  1      2  2      2  3 

•*•       5  5     5'5  "5   •  **•       3  5  "3  ?     3   •  J-^-     -7  5     7^5  ~7   • 

A       1-8     li>     _2_Q.  Q      12      IJl     14  lA       1«     li)      2_0 

*•       9  5     9"?     9  •  ^-        6  5  '6   5     6*  •'■*•     '6  5     6"5  '6   ' 

c       2Jb     2_5     2_6  in      JL4     15     16  ,«       15     16     17 

S-     "8'"5     85  -8--  10.     -S7-,  -y-,  -y-.  15.     -J-,  -5--,  -5-. 

189.  Written  Exercises. 
1.   Multiply  $6.50  by  3. 

Model  :      $  6.50         Multiply  as  in  previous  exer- 

3     cises,  and  point  off  two  places 

$  19.50     for  cents. 

2.  $656.50x2  7.  $897.68x2  12.  $302.23x7 

3.  $329.40x3  8.  $950.75x3  13.  $310.32x8 

4.  $345.54x4  9.  $432.50x4  i4.  $231.12x9 

5.  $453.45x5  10.  $345.24x5  15.  $330.30x9 

6.  $323.10x6  11.   $230.20x6  16.  $103.23x8 

190.  Written  Problems. 

Give  the  analysis  for  each  : 

1.  At  $4.75  eachj  what  will  be  the  cost  of  3  sheep  ? 

2.  What  will  be  the  cost  of  4  chairs  at  $3.25  each? 

3.  At  $42.50  each,  what  will  be  the  cost  of  4  cows? 

4.  A  boy  earns  $15.75  a  month.     How  much  will 
he  earn  in  3  months  ? 


136  MULTIPLICATION   AND    DIVISION 

5.  How  much  will  3  railroad  tickets  from  Chicago 
to  San  Francisco  cost  at  $62.50  each? 

6.  A  man  pays  $  13.50  a  month  rent  for  a  house. 
How  much  will  this  amount  to  in  4  months? 

7.  Find  the  cost  of  3  tons  of  coal  at  $6.75  a  ton. 

8.  A  man  bought  sheep  at   $4  each.     He  paid 
$176  for  the  sheep.     How  many  sheep  did  lie  buy? 

9.  What  is  the  unit  of  measure  in  Problem  8  ? 

10.  A  man  owned  360  acres  of  land.  He  sold  ^  of 
it.  How  many  acres  did  he  sell?  How  many  acres 
did  he  have  left  ? 

191.    Oral  Exercises. 

Add  as  indicated  in  Step  C,  p.  49. 


a 

6 

C 

d 

e 

X 

9 

h 

i 

J 

Z: 

I 

m 

n 

2 

4 

2 

3 

3 

3 

3 

9 

2 

4 

2 

9 

1 

5 

3 

5 

2 

4 

3 

r 
0 

6 

5 

4 

4 

7 

1 

3 

4 

5 

8 

7 

5 

4 

2 

1 

3 

4 

2 

1 

4 

6 

8 

2 

3 

9 

8 

3 

3 

3 

2 

2 

4 

2 

5 

1 

3 

3 

4 

2 

3 

3 

5 

6 

5 

4 

4 

7 

1 

3 

5 

8 

5 

2 

4 

5 

8 

4 

3 

7 

2 

6 

4 

8 

4 

3 

3 

4 

8 

9 

4 

8 

8 

6 

9 

8 

9 

6 

7 

!) 

7 

6 

2 

6 

8 

8 

9 

7 

9 

6 

1 

6 

3 

Whenever  two  numbers  whose  sum  is  not  more 
than  9  are  to  be  added  to  10,  20,  30,  etc.,  take  both 
numbers  together. 

Exercise  a  above  may  be  added:  12,  20,  25,  30,  35. 
Add  the  above  exercises  in  a  similar  manner. 


MULTIPLICATION  — LESSON  F  137 

MULTIPLICATION  — LESSON  F 

192.     1.    Memorize  the  following  : 

6  7  6  4  4  5  6 

x4  x4  x5         x7         x6         x6  x6 

"24         "28  30  28  24  30  36 

2.  Count  by  5's  to  30 ;  by  6's  to  36. 

3.  Count  by  4's  to  28  ;  by  7's  to  28. 

4.  There  are  7  days  in  one  week.  How  many 
days  are  there  in  4  weeks  ? 

5.  There  are  4  quarts  in  one  gallon.  How  many 
quarts  are  there  in  six  gallons  ? 

6.  What  is  the  product  of  4  and  7  ?     Of  4  and  6  ? 

7.  Alice  bought  6  pencils  at  4^  each.  She  handed 
the  clerk  a  25-cent  piece.  How  much  change  should 
she  receive  ? 

8.  A  boy  sold  6  papers  at  5^  each.  How  much 
money  did  he  receive  for  all  ? 

9.  At  $4  each,  what  will  be  the  cost  of  7  chairs  ? 

10.  What  will  be  the  cost  of  6  tablets  at  6^  each? 

11.  At  the  rate  of  5  marbles  for  a  cent,  how  many 
marbles  can  be  bought  for  6  ^  ? 

12.  A  girl  bought  4  yd.  of  ribbon  at  6^  a  yard. 
She  gave  the  clerk  25  cents.  How  much  change 
should  she  receive  ? 

13.  How  many  cents  are  6  nickels  ?  How  many 
dollars  are  six  5-dollar  gold  pieces  ? 


138  MULTIPLICATION  AND  DIVISION 

193.    Study  Exercises. 

6  7  6  4  4  5  6 

x4         x4         x5         x7         x6         x6         x6 

Study  the  above  exercises  until  you  can  give  the 
products  without  hesitation. 

Give  the  products  from  right  to  left,  adding  the 
following  numbers  to  each  product :  3,  4,  7,  8. 

.194.     Multiply: 

1.  677,676  X  34  ii.  654,564  x  56  21.  321,213  x  78 

2.  765,756  X  24  12.  365,456  x  56  22.  203,320  x  68 

3.  456,746  X  14  13.  246,365  x  65  23.  332,223  x  58 

4.  375,647  X  34  i4.  654,321  x  56  24.  123,123  x  48 

5.  263,746  X  24  15.  506,430  x  46  25.  301,203  x  38 

6.  565,656  x  54  le.  434,343  x  67  26.  332,233  x  89 

7.  456,546  X  45  17.  342,434  x  57  27.  312,013  x  79 

8.  346,543  X  54  la  234,342  x  47  28.  120,320  x  69 

9.  425,636  X  35  19.  324,130  x  37  29.  231,032  x  59 
10.  654,321  X  45  20.  123,432  x  27  30.  123,320  x  49 

195.  Solve: 

$913.78         $935.36         $8312.75  $835.00 

-$535.79      -$145.68      -$2353.76      -$135.25 

What  is  the  answer  in  division  called  ? 


DIVISION  — LESSON   F  139 

DIVISION  — LESSON  F 

196.     1.    Memorize  the  folloiving  : 

6  '^  _5         _i         _i         __^         _^ 

4)24       4)28       5)30       7)28       6)24       6)30       6)36 

2.  How  many  weeks  are  there  in  28  days  ? 

3.  How  many  gallons  are  there  in  24  quarts  ? 

4.  How  many  nickels  are  there  in  30  cents  ? 

5.  What  is  J  of  $24?     What  is  i  of  $36  ? 

6.  $6isiof$— .     $7isiof$— . 

7.  At  5^  each,  how  many  oranges  can  be  bought 
for  30^? 

8.  At  $4  each,  how  many  chairs  can  be  bought 
for  $28? 

9.  A  girl  had  30  cents.     She  spent  ^  of  her  money 
for  a  tablet.     What  was  the  cost  of  the  tablet  ? 

10.  How  many  pounds  of  sugar  at  6  ^  a  pound  will 
cost  24^? 

11.  What  is  the  unit  of  measure  in  Problem  8  ? 

12.  There  are  36  pupils  in  a  schoolroom.  There 
are  6  pupils  seated  in  each  row.  How  many  rows  of 
seats  are  there  in  the  room  ? 

13.  If  each  stove  costs  $  7,  how  many  stoves  can 
be  bought  for  $28? 

14.  If  a  boy  earns  $  6  each  month,  how  long  will 
it  take  him  to  earn  $24  ? 

15.  What  is  the  ratio  of  6  to  24  ?     Of  24  to  6  ? 


140  MULTIPLICATION   AND  DIVISION 

197.  Study  Exercises. 

4)24  4)28       5)30       7)28  6)24  6)30  6)36 

4)25  4)29       5)3T       7)29  6)2'5  6)3"l  6)37 

4)26  4)30       5)32       7)30  6)"26  6)32  6)38 

4)27  4)31       5)33       7)"3l  7)27  6)T3  6)39 

Study  the  above  exercises  until  you  can  give  the 
quotients  without  hesitation. 

Give  the  quotients  and  the  remainders,  if  any. 

Give  the  quotients  with  the  remainders  expressed 
as  fractions. 

198.  Written  Exercises. 

Divide  the  numbers  in  each  cohimn  by  the  numbers 
above  the  columns. 


4,3 

5,2 

6,3 

7,2 

15-242,824 

11.  303,305 

21. 

243,024 

31.  212,821 

2A268,264 

12.  252,030 

22. 

254,544 

32.  296,968 

35  264,268 
4.''806,704 
s.^307,048 

13.  318,180 

23. 

266,736 

33.  233,814 

14.  323,230 

24. 

393,936 

34.  226,257 

15.  272,780 

25. 

■  272,780 

35.  309,401 

6.  257,770 

16.  267,605 

26. 

267,606 

36.  156,268 

7.  226,570 

17.  813,215 

27. 

813,215 

37.  870,401 

a.  270,264 

18.  627,110 

28. 

627,120 

38.  994,714 

9.  936,536 

19.  758,230 

29. 

758,130 

39.  169,547 

10.  623,012 

20.  126,315 

30. 

126,315 

40.  714,931 

DRILL   EXERCISES 


141 


199.    Drill  Exercises. 

Give  quotients  with  the  remainders  expressed  as 


fractions : 

1-  ¥,  ¥.  ¥• 

7. 

¥.  ¥.  ¥• 

13. 

¥, 

¥> 

¥• 

2.  ¥,  -¥->  -¥■ 

8. 

¥.  ¥.  ¥- 

14. 

¥. 

¥. 

¥• 

3.  V,  ¥.  ¥• 

9. 

¥.  ¥,  V- 

15. 

¥, 

¥, 

18 
4  • 

4-  ¥'  ¥>  ¥• 

10. 

¥>  ¥.  ¥• 

16. 

¥- 

¥> 

¥• 

5-  ¥.  ¥.  -V- 

11. 

¥.  ¥>  ¥• 

'  17. 

¥, 

1  7 

¥• 

6.  -¥  ¥.  ¥- 

12. 

¥,  ¥>  ¥• 

18. 

¥> 

¥, 

¥■ 

200.    Oral  Exercises. 

Add  each  cohnnii  as  indicated  in  Step  C,  p.  49. 


a 

b 

C 

d 

e 

/ 

</ 

/( 

i 

J 

A: 

I 

m 

3 

5 

2 

6 

6 

8 

6 

6 

6 

9 

6 

9 

5 

8 

7 

9 

7 

6 

8 

6 

1^ 
1 

9 

i 

5 

8 

7 

9 

8 

9 

7 

8 

4 

8 

7 

5 

4 

9 

3 

8 

3 

5 

2 

6 

6 

8 

6 

6 

6 

9 

5 

9 

5 

8 

7 

9 

7 

C 

8 

6 

7 

9 

7 

6 

8 

8 

9 

8 

9 

7 

8 

7 

8 

5 

5 

3 

8 

5 

7 

4 

7 

2 

1 

5 

9 

3 

7 

8 

6 

8 

6 

6 

3 

8 

9 

5 

4 

9 

2 

8 

9 

6 

7 

4 

Whenever  any  two  number.^  are  to  be  added  to  10, 
20,  30,  etc.,  take  both  numbers  together. 

Exercise  a  above  may  be  added  :  10,  27,  30,  47,  50. 
Exercise  h  above  may  be  added  :   10,  25,  30,  45,  50. 
Add  the  above  exercises  in  a  similar  manner. 
Add  the  above  exercises  from  the  top  down. 


142  MULTIPLICATION  AND  DIVISION 

201.    Oral  Exercises. 

Review  Exercises  90  and  130. 

1.  If  a  pie  is  cut  into  thirds,  it   is  cut  into  — 
equal  pieces. 

2.  A  woman  gave  each  hoj  .^  of  a  pie.     She  gave 
the  boys  2  pies.     How  many  boys  were  there  ? 

3.  A  woman  gave  each  of  9  boys  ^  of  a  pie.     She 
gave  the  boys  —  pies. 

4.  A  woman  gave  each  boy  ^  of  a  pie.     It  took 
3 1  pies.     There  were  —  boys. 

5.  A  woman  divided  two  pies  equally  among  6 
boys.     What  part  of  a  pie  did  each  boy  receive  ? 

6.  In  one  pie  there  are  —  thirds  of  a  pie. 

7.  In  2  apples  there  are  —  thirds  of  an  apple. 
jB.    2 J  apples  =  f  apples.     3 J  feet  =  |  feet. 

9.  ^-  inches  =  —  inches.     ^^-  inches  =  —  ^  inches. 

10.  -^3^-  pies  =  —  pies.     ^  apples  =  —  -|  apples. 

11.  41  yards  =  f  yards.     |  days  =  —  i  days. 

12.  f  apples  =  —  apples.     ^-  pies  =  —  ^  pies. 

13.  5 1  apples  and  2^  apples  are  —  apples. 

14.  3 J  yards  and  2^  yards  are  —  yards. 

15.  2^  pies  and  If  pies  are  —  pies. 

16.  5^  feet  and  2f  feet  are  —  feet. 


MULTIPLICATION  — LESSON   G  143 

MULTIPLICATION— LESSON  G 
202.     1.    Memorize  the  foUotving  : 

8  9  7  4  4  5 

x4        x4         x5         x8         x9         x7 

32         36  35         ^  36         ^ 

2.  How  many  days  are  there  in  5  weeks  ? 

3.  Nine  gallons  are  how  many  quarts  ? 

4.  At  8^  a  yard,  what  will  be  the  cost  of  4  yards 
of  ribbon  ? 

5.  There  are  8  quarts  in  one  peck.     How  many 
quarts  are  there  in  4  pecks  ? 

6.  Seven  nickels  are  how  many  cents  ? 

7.  How  much  will  7  bars  of  soap  cost  at  5^^  each? 

8.  At  $  9  a  ton,  how  much  will  4  tons  of  coal 
cost  ? 

9.-  How  much  will  4  boxes  of  berries  cost  at  8^  a 
box  ? 

10.  Count  by  4's  to  36  ;  by  5's  to  35  ;  by  8's  to  32. 

11.  At  the  rate  of  9  miles  an  hour,  how  far  will  a 
boat  sail  in  4  hours  ? 

12.  If  there  are  5  rows  of  seats  in  a  schoolroom, 
and  7  seats  in  each  row,  how  many  seats  are  there  in 
the  room  ? 

13.  A  boy  delivers  5  quarts  of  milk  each  day. 
How  many  quarts  will  he  deliver  in  one  week? 


144  MULTIPLICATION   AND   DIVISION 


203.    Study  Exercises. 

8          9           7 

4 

4 

5 

X  4       X  4        X  5' 

x8 

x9 

x7 

Study  the  above  until  you  can  give  the  products 
without  hesitation. 

Give  the  products  from  right  to  left,  adding  the 
following  to  each  product:  6,  4,  7,  9. 

204.  Written  Exercises. 

1.  898,989x34  7.  234,324x89  13.  543,235x75 

2.  989,898x34  a  324,432x89  i4.  676,767x45 

3.  696,896x34  9.  432,342x89  15.  456,657x45 

4.  789,987x34  10.  123,432x89  le.  765,432x45 

5.  579,869x34  11.  343,210x89  17.  123,567x25 

6.  456,789x34  12.  543,345x67  la  345,765x25 

205.  Oral  Exercises. 

Supply  the  number  in  place  of  x. 

1.  6x6  =  9x^  9.  4x9  =  6  X  a; 

.*2.  6x4  =  8xx  10.  5x4  =  2xa; 

3.  6x5  =  5xic  11.  8x3  =  4xa; 

4.  8x2  =  4x03  12.  5x6  =  6xa; 

5.  2x9  =  6xa;  13.  4x4  =  2xx 

6.  3x5  =  5xx  14.  4x6  =  8xa; 

7.  2x6  =  4xa:  15.  6x5  =  3xa: 

8.  3x6  =  9xa;  I6.  6x3  =  2xa: 


DIVISION  — LESSON   G  145 

DIVISION  — LESSON  G 
206.    1.    Memorize  the  folloiving : 

_8  __9  _7  ^  _4  ^ 

4)32  4)36  5)3'5  8)32  9)36  7)35 

2.  At  5^  each,  how  many  oranges  can  be  bought 
for  35y  ? 

3.  What  part  of  $  36  is  $  9  ?     What  is  the  ratio 
of  $  9  to  $  36  ? 

4.  If  9  chairs  cost  %  36,  what  will  be  the  cost  of  1 
chair  ? 

5.  If  a  horse  is  fed  9  quarts  of  oats  a  day,  how 
many  days  will  3.6  quarts  last  ? 

6.  What  is  the  unit  of  measure  in  Problem  5  ? 

7.  There  are  8  pints    in    1    gallon.     How  many 
gallons  are  there  in  32  pints  ? 

8.  If  a  horse  travels  36  miles  in  9  hours,  what  is 
the  average  rate  per  hour  ? 

9.  If  7  boys  together  have  35  marbles,  what  is 
the  average  number  for  each  boy  ? 

10.  Thirty-six  boxes  of  berries  \vere  picked  from  a 
garden  in  4  days.  What  was  the  average  number  oi 
boxes  for  each  day  ? 

11.  The  number  of  boxes  of  berries  picked  from  a 
garden  averages  4  each  day.  How  many  boxes  will 
be  picked  in  8  days  ? 

12.  If  a  boy  rides  his  wheel  at  an  average  rate  of 
5  miles  an  hour,  how  far  will  he  ride  in  9  hours  ? 

*  1st  Bk  A  KIT  1 1 — 10 


146 

MULTIPLICATION 

AND   DIVISION 

207. 

Study  : 

Exercises. 

4)32 

4)36 

5)35 

8)32 

9)36 

7)35 

4)33 

4)37 

5)36 

8)33 

9)37 

7)36 

4)34 

4)38 

5)37 

8)34 

9)38 

7)37 

4)35 

4)39 

5)38 

8)35  _ 

9)39 

7)38 

Give  the  quotients  and  the  remainders,  if  any. 

Give  the  quotients  with  the  remainders  expressed 
as  fractions. 

With  4  as  a  divisor,  what  is  the  largest  remainder 
possible  ?     Why  ? 

What  is  the  largest  remainder  possible  with  3  as  a 
divisor  ? 

If  the  quotient  is  3 J,  can  you  tell  what  the  divi- 
sor was? 

208.    Written  Exercises. 

Divide  the  numbers  in  each  column  by  the  numbers 
above  the  column. 

4,  3,  2  ■        5,  2,  3             7,  4,  3             8,  4,  2 

1.  323,632  6.  388,385  ii.  352,821  is.  322,416 

2.  356,356  7.  378,270  12.  387,471  17.  339,536 

3.  339,392  8.  232,375  13.  248,780  is.  275,464 

4.  343,436  9.  183,180  i4.  283,115  19.  984,804 

5.  387,872  10.  867,325  is.  239,708  20.  115,304 


ORAL  EXERCISES  147 

209.    Oral  Exercises. 

1.  How  can  you  find  |  of  a  number  ?  J  of  a  number  ? 

2.  What  is  i  of  4  ?      Of  6  ?      Of  8  ?      Of  12  ? 
Of  7  ?     Of  5  ?     Of  9  ?     Of  19  ?     Of  24  ? 

3.  What  is  1  of  6  ?      Of  12  ?      Of  7  ?      Of  8  ? 
Of  10?     Of  11?     Of  13?     Of  14?     Of  9? 

4.  When  you  know  what  ^  of  a  number  is,  how 
can  you  find  what  the  number  is  ? 

5.  When  you  know  what  ^  of  a  number  is,  how 
can  you  find  what  that  number  is  ? 

6.  When  you  know  what  i  of  a  number  is,  how 
can  you  find  |  of  the  number  ? 

7.  Find  i  of  12.  Find  |  of  12.  Find  f  of  9. 
a  If  1^  of  a  number  is  3,  what  is  the  number  ? 
9.    If  ^  of  a  number  is  5,  what  is  the  number  ? 

10.  If  J  of  a  number  is  4,  what  is  the  number  ? 

11.  If  I  of  a  number  is  4,  what  is  ^  of  the  number  ? 

12.  If  I  of  a  number  is  4,  what  is  the  number  ? 

13.  If  I  of  a  number  is  6,  what  is  the  number  ? 

14.  If  I  of  a  number  is  2,  what  is  ^  of  the  number  ? 

15.  What  is  f  of  6  ?     Of  9  ?     Of  12  ?     Of  3  ? 

16.  What  is  1  of  1?     Of  2?     Of  3?    Of  4?    Of  5? 


17.  If  ^  of  the  number  of  marbles  a  boy  has  is  6, 
lat  are  |  of  the  number  of  marbles  ? 

18.  If  f  of  the  number  of  books  a  girl  has  are  6, 
lat  is  ^  of  the  nui 

many  books  has  she  ? 


what  is  ^  of  the  number  of  books  she  has  ?     How 


148  MULTIPLICATION    AXD   DIVISION 

210.    Oral  Problems.* 

1.    What  are  f  of  $15? 

Model  for  oral  recitation:  Since  ^  of  $15  is  $5, 
I  of  $15  are  2  times  $5,  or  $10. 
.    Model  for  written  recitation  : 

$5  is  1  of  $15.  $5  is  1  of  $15. 


3)$  15  j<^ 

$10  are  f  of  $15. 

2.  What  are  f  of  $  18  ?     Of  $  24  ?     Of  15  days  ? 

3.  What  are  |  of  21  days  ?     Of  12^?     Of  15  ft.  ? 

4.  What  are  |  of  $  16  ?     Of  12  days  ?     Of  $  24  ? 

5.  What  are  |  of  $15  ?     Of  $20  ?     Of  25^? 

6.  At  18^  a  yard,  how  much  will  f  of  a  yard  of 
cloth  cost? 

7.  There  are  12  months  in  1  year.     How  many 
months  are  there  in  J  of  a  year  ? 

8.  flow  many  inches  are  there  in  f  of  a  foot  ? 

9.  How  many  pints  are  there  in  |-  of  a  gallon  ? 

10.  A  boy  had  16  marbles.  He  sold  ^  of  them. 
How  many  marbles  did  he  sell  ? 

11.  A  girl  bought  f  of  a  yard  of  ribbon  at  20^  a 
yard.     How  nuich  did  the  ribbon  cost  her  ? 

12.  A  boy  had  12  miles  to  travel.  He  rode  |  of 
the  distance  and  walked  the  remainder.  How  far 
did  he  ride  ?     How  far  did  he  walk  ? 

*  See  note,  p.  116. 


WRITTEN   PROBLEMS  149 

211.   Written  Problems. 

1.  Find  the  cost  of  19  cows  at  $34  each. 

2.  At  $7  a  ton,  how  many  tons  of  coal  can  be 
bought  for  $224? 

3.  Find  the  number  of  square  rods  in  a  field  43 
rods  long  and  29  lods  wide. 

4.  If  a  train  travels  at  an  average  rate  of  45  mi. 
an  hour,  how  far  will  it  travel  in  one  day  ? 

5.  A  boat  traveled  3150  miles  in  one  week.  What 
was  the  average  distance  traveled  each  day  ? 

6.  A  fanner  raised  32  bu.  of  oats  to  the  acre. 
How  many  bushels  did  he  raise  from  28  acres  ? 

1.  A  boy  picked  78  boxes  of  apples  in  6  days. 
What  was  the  average  number  of  boxes  picked  each 
day? 

8.  There  are  60  seconds  in  one  minute  and  60 
minutes  in  one  hour.  How  many  seconds  are  there 
in  one  hour  ? 

9.  How  many  minutes  are  there  in  one  day  ? 

10.  How  many  weeks  are  there  in  364  days  ? 

11.  A  man  sold  hay  at  $8  a  ton.     He  received 
$96  for  it.     How  many  tons  did  he  sell? 

12.  How  much  is  hay  worth  a  ton  when   $  72  is 
paid  for  6  tons  ? 

13.  How  many  days  are  there  in  14  weeks  ? 

14.  Find  the  value  of  4  carloads  of  coal,  each  con- 
taining 16  tons,  at  $6  a  ton. 


150  MULTIPLICATION^  AND  DIVISION 

212.   Oral  Problems.* 

1.  If  $15  is  f  of  the  cost  of  a  stove,  what  is  the 
cost  of  the  stove  ? 

Model  for  oral  recitation  :  If  $15  is  |  of  the  cost 
of  the  stove,  |  of  the  cost  of  the  stove  is  ^  of  $15, 
or  $  5.  If  $  5  is  i  of  the  cost  of  the  stove,  the  cost 
of  the  stove  is  4  times  $  5,  or  $  20. 

Model  for  written  recitation  : 

$5  is  1  of  the  cost.  $ 5  is  i  of  the  cost. 

3)$  15  is  I  of  the  cost.  _x4 

$20  is  I  of  the  cost. 

2.  If  $12  is  f  of  the  cost  of  a  suit  of  clothes,  what 
is  the  cost  of  the  suit  ? 

3.  If  $20  is  1^  of  the  cost  of  a  cow,  what  is  the 
cost  of  the  cow  ? 

4.  If  f  of  the  cost  of  a  book  is  18^',  what  is  the 
cost  of  the  book  ? 

5.  If  f  of  the  cost  of  a  cap  is  24^,  what  is  the  cost 
of  the  cap  ? 

6.  A  boy  sold  f  of  his  marbles  for  14^.  At  the 
same  rate,  how  much  were  all  of  his  marbles  worth  ? 

7.  If  f  yd.  of  cloth  is  worth  21^,  how  much  is 
1  yd.  worth  ? 

8.  A  girl  spent  9  weeks  visiting  her  aunt.  This 
was  J  of  her  vacation.     How  long  was  her  vacation  ? 

9.  Fred's  age  is  12  years.  He  is  |  as  old  as  Harry. 
How  old  is  Harry  ? 

*  See  note,  p.  115. 


5 

6 

x9 
45 

x7 
42 

MULTIPLICATION  — LESSON  H  151 

MULTIPLICATION  — LESSON   H 

213.  1.    Memorize  the  foUoiving  : 

8  9           7           7           5 

X  5  X  5        x_6        x_7        x8 

"40  l5        l2        19        To 

2.  Count  by  5's  to  45 ;  by  9's  to  45. 

3.  Count  by  6's  to  42  ;  by  7's  to  49. 

4.  How  many  days  are  there  in  6  weeks  ?    In  7 
weeks  ? 

5.  Eight  nickels  are  how  many  cents  ? 

6.  Nine  nickels  are  how  many  cents  ? 

7.  Count  by  8's  to  40;  by  4's  to  40. 

8.  What  will  be  the  cost  of  6  clocks  at  1 7  each  ? 

9.  If  oranges  are  worth  5)^  each,  what  will  be  the 
cost  of  9  oranges  ? 

10.  What  will  be  the  cost  of  8  loaves  of  bread  at 
5^  a  loaf? 

11.  In  preparing  for  a  party,  some  girls  bought 
7  boxes  of  berries  at  6^  each.  They  handed  the 
dealer  a  half-dollar.  How  much  change  should  they 
receive  ? 

12.  In  planting^his  orchard,  a  farmer  set  out  7  rows 
of  trees  with  7  trees  in  each  row.  Make  a  drawing  to 
show  the  trees.     How  many  trees  did  he  set  out  ? 

13.  How  many  square  inches  are  there  in  an  oblong 
7  in.  long  and  6  in.  wide  ?  Make  a  drawing  to  show 
this. 


152  MULTIPLICATION   AND   DIVISION 


214.  Study  Exercises. 

8    9    7    7 

5 

5 

6 

x5   x5   x6   x7 

x8 

x9 

x7 

Study  tbe  above  exercises  until  you  can  give  the 
products  without  hesitation. 

Give  the  products  from  right  to  left,  adding  the 
follov^ring  to  each  product :  6,  7,  8,  9. 


215.    Written  Exercises. 


1. 

897,879  X  45 

11. 

765,675  X  67 

2. 

789,789x45 

12. 

567,576  X  67 

3. 

687,969  X  45 

13. 

657,756  X  67 

4. 

987,654x45 

14. 

456,767  X  67 

5. 

456,789  x  45 

15. 

765,432  X  67 

6. 

978,645  X  35 

16. 

543,435x89 

7. 

697,548  X  35 

17. 

435,534  X  89 

8. 

569,784  X  35 

18. 

234,530  X  89 

9. 

678,945  X  25 

19. 

524,321x89 

LO. 

987,456  X  25 

20. 

454,302x89 

216.    Write  in  a  column  and  add : 

1.  $375.40,  $1087.09,  $9.75,  $84.75,  $450,  $90 

2.  $  90,  $  100.25,  $  .90,  $  1 .75,  $  3750,  $  80.80,  $  90 

3.  $3.75,  $9375,  $.97,  $105,  $7.25,  $725,  $8.00 

4.  $.95,  $95,  $9.50,  $950,  $9500,  $90.50,  $5.95 

5.  $1.75,  $175,  $17.50,  $1750,  $175.50,  $17.05 


DIVISION  — LESSON   H  153 

DIVISION -LESSON  H 
217.    1.    Memorize  the  following : 

_8        _9        J7        _J7        _5        _5       _6 
5)40      5)45      6)42      7)49      8)40      9)45      7)42 

2.  What  part  of  40  is  5  ?     What  part  of  40  is  8  ? 

3.  What  part  of  $45  is  $9?     What  part  of  $45 
is  $5? 

4.  What  is  the  ratio  of  $  5  to  $  40  ?    Of  $  40  to  $  5  ? 

5.  A  girl  spent  6   weeks  in  the  country.     How 
many  days  did  she  spend  in  the  country  ? 

6.  A  boy  spent  49  days  in  the  city.     How  many 
weeks  did  he  spend  in  the  city  ? 

7.  A  boy  sold  papers  at  5^  each.     He  received 
45^.     How  many  papers  did  he  sell  ? 

8.  A  girl  receives  5^  a  day  for  helping  her  aunt. 
In  how  many  days  will  she  earn  40^? 

9.  A  girl  picked  G  boxes  of  berries  and  sold  them 
for  7^  a  box.     How  much  did  she  receive  for  them? 

10.  A  farmer  set  out  45   apple  trees   in  5  rows. 
How  many  trees  did  he  set  out  in  each  row  ? 

11.  There  are  42  square  inches  in  an  oblong.     The 
oblong  is  7  inches  long.     How  wide  is  it? 

12.  There  are  40  square  inches  in  an  oblong.     The 
oblong  is  5  inches  wide.     How  long  is  it  ? 

13.  A  man  spent  45^  for  car  fares.     He  paid  5^  for 
each  ride.     How  many  times  did  he  ride  ? 


154  MULTIPLICATION  AND  DIVISION 

218.   Study  Exercises. 

5)40      5)45      6)42      7)49      8)40      9)45      7)42 


5)41 

5)46 

6)43 

7)50 

8)41 

9)46 

7)43 

5)42 

5)47 

6)44 

7)51 

8)42 

9)47 

7)44 

5)43 

5)48 

6)45 

7)52 

8)43 

9)48 

7)45 

5)44      5)49      6)46      7)53      8)44      9)49      7)46 

Study  the  above  exercises  until  you  can  give  the 
quotients  without  hesitation. 

Give  the  quotients  with  the  remainders  expressed 
as  fractions. 

219.    Written  Exercises. 

Divide  the  numbers  in  each  column  by  the  numbers 


above  the  column. 

5,  2 

6,  4 

7,  3 

8,  4 

1. 

404,535 

9. 

423,660 

17. 

354,228 

25. 

403,224 

2. 

449,885 

10. 

364,042 

18. 

465,962 

26. 

434,416 

3. 

439,380 

11. 

460,632 

19. 

459,585 

27. 

443,640 

4. 

429,340 

12. 

448,464 

20. 

394,954 

28. 

436,280 

5 

287,965 

13. 

183,936 

21. 

325,255 

29. 

347,560 

6. 

478,470 

14. 

225,870 

22. 

304,038 

30. 

178,024 

7. 

198,395 

15. 

166,596 

23. 

254,422 

31. 

188,000 

8. 

973,285 

16. 

103,032 

24. 

179,282 

32. 

202,000 

220.    Solve: 

a 

6 

c 

d 

$105.25 

1374.68 

$860.50 

$334.57 

-$55.50 

- 

$160.69 

— 

$292.68 

-$85.79 

ORAL  PROBLEMS  155 

221.    Oral  Problems.* 

1.  How  many  sheep  at  $  6  each  must  a  farmer  sell 
to  pay  for  3  tons  of  hay  at  $  10  a  ton  ? 

Model  for  oral  recitation :  The  cost  of  3  tons  of 
hay  at  $10  a  ton  is  $30.  To  receive  $30,  the 
farmer  must  sell  as  many  sheep  as  $6  is  contained 
in  $30,  or  5.     He  must  sell  5  sheep. 

Model  for  written  recitation : 

$  10,  cost  of  1  ton. 

x3 
$  30,  cost  of  3  tons. 

5,  number  of  sheep, 
price  of  1  sheep,  $6)$30 

2.  At  $  4  each,  ho,w  many  sheep  must  a  farmer  sell 
to  pay  for  3  tons  of  coal  at  $  8  a  ton  ? 

3.  How  many  boxes  of  berries  at  8^  a  box  must  a 
boy  sell  to  pay  for  4  tablets  at  10^  each  ? 

4.  How  many  pounds  of  sugar  at  6^  a  pound 
should  a  woman  receive  for  3  doz.  eggs  at  10^  a 
dozen  ? 

5.  At  8^  a  pound,  how  many  pounds  of  raisins 
should  be  exchanged  for  2  lb.  of  butter  at  40^  a 
pound  ? 

6.  A  girl  bought  4  yd.  of  ribbon  at  10^  a  yard. 
She  paid  for  it  in  berries  at  8^  a  box.  Find  the 
number  of  boxes  required, 

*  See  note,  p.  115. 


156  MULTIPLICATION   AND  DIVISION 

222.   Oral  Exercises. 

1.  How  many  fourths  are  there  in  1  apple  ?     In  2 
apples  ?     In  one  dollar  ?     In  2  dollars  ? 

2.  How  many  fourths  are  there  in  2]-  dollars  ?   In 
3  J  pies  ?     In  1|  circles  ?     In  5f  dollars  ? 

3.  A  woman  gave  each  of  12  boys  ^  of  a  pie. 
How  many  pies  did  it  take  ? 

4.  A  man  gave  each  boy  ^  of  a  dollar.     He  gave 
the  boys  2J  dollars.     How  many  boys  were  there  ? 

5.  In  J  of  a  pie  there  are  —  fourths  of  a  pie. 

6.  In  1^  of  a  pie  and  J  of  a  pie  there  are  —  fourths 
of  a  pie. 

7.  Three  dollars  are  how  many  quarter  dollars  ? 

8.  If  J  of  the  number  of  boys  in  a  class  is  2,  how 
many  boys  are  there  in  the  class  ? 

9.  If  J  of  the  number  of  girls  in  a  room  is  6,  how 
many  girls  are  there  in  the  room  ? 

10.  When  you  know  what  J  of  a  number  is,  how 
do  you  find  what  the  number  is  ? 

11.  When  you  know  what  J  of  a  number  is,  how 
do  you  find  what  ^  of  the  number  is  ? 

12.  When  you  know  what  f  of  a  nimaber  is,  how 
do  you  find  what  ^  of  tlie  number  is  ? 

13.  There  are  16  girls  in  a  class.  Find  J  of  the 
number  of  girls  there  are  in  the  class.  Find  J  of  the 
number  of  girls  there  are  in  the  class. 


ORAL  EXERCISES  167 

223.  Oral  Exercises. 

1.  If  an  apple  is  cut  into  5  equal  parts,  what  is 
one  of  the  parts  called  ? 

2.  In  one  apple  there  are  —  fifths  of  an  apple.    In 
2  apples  there  are  —  fifths  of  an  apple. 

3.  In  2|  apples  there  are  —  fifths  apples.     In  3J- 
apples  there  are  —  fifths  apples. 

4.  In  3|-  apples  there  are  f  apples.     In  2^  apples 
there  are  f  apples.     In  4^  apples  there  are  f  apples. 


5. 

In  4  there  are  f .    In  5  there  are  |.     In  4  there 

a^e  f , 

6 

In  3  there  are  f .    In  5  there  are  |.    In  6  there 

are  f, 

7. 
8. 

9. 

The  ratio  of  1  to  5  is  — .    The  ratio  of  5  to  1  is  — . 

10. 

The  ratio  of  1  to  ^  is  — .    The  ratio  of  |^  to  1  is  — . 

11.  How  many  3-inch  rulers  will  a  12-inch  ruler 
make  ? 

12.  How  many  units  of  ^  ft.  are  there  in  1  ft.? 

13.  In  1  ft.  there  are  how  many  units  of  l  ft.  ?     Of 
1ft.?     Of  J  ft.? 

14.  Express  as  w^hole  numbers  :  J^  in. ;  ^  ft. 

15.  Express  as  thirds  :  21  in.;  4|  ft.;  5 J  hr.;  6|  da. 

16.  If  a  circle  is  cut  into  6  equal  parts,  what  is  one 
of  the  parts  called  ? 

17.  Which  is  the  largest  fraction  of  a  circle :  J,  ^, 
or  1  of  a  circle  ? 


158  MULTIPLICATION   AND  DIVISION 

224.   Oral  Problems.* 

1.  If  4  chairs  cost  $  16,  how  much  will  3  chairs 
cost  ? 

Model  for  oral  recitation:  If  4  chairs  cost  $16,  1 
chair  will  cost  ^  of  $16,  or  $4.  Since  1  chair  costs 
$4,  3  chairs  will  cost  3  times  $4,  or  $12. 

Model  for  written  recitation : 

$4,  cost  of  1  chair.  $4,  cost  of  1  chair. 


16,  cost  of  4  chairs.  x  3 

$  12,  cost  of  3  chairs. 

2.  If  4  hats  cost  $  8,  how  much  will  5  hats  cost  ? 

3.  If  3  tables  cost  $18,  how  much  will  5  tables 
cost? 

4.  If  6  tablets  cost  24)^,  how  much  will  4  tablets 
cost? 

5.  If  3  lb.  of  sugar  cost  18^,  how  much  will  2  lb. 
cost? 

6.  At  the  rate  of  4  for  20^,  how  much  will  7  bars 
of  soap  cost  ? 

7.  A  girl  bought  4  yd.  of  ribbon  for  24^.  How 
much  would  6  yd.  have  cost  her  at  the  same  rate  ? 

8.  A  boy  picked  6  boxes  of  berries.  He  sold  2 
boxes  for  14^.  How  much  would  he  have  received 
for  6  boxes  at  the  same  rate  ? 

9.  George  rode  12  mi.  in  2  hours.  At  the  same 
rate,  how  far  would  he  ride  in  5  hours  ? 

*  See  note,  p.  115. 


MULTIPLICATION  — LESSON  I  159 

MULTIPLICATION— LESSON   I 
225.   1.   Memorize  the  folloioing : 
8  9  8  6  6  ,7 

x6  x6  x7  x8  x9  x8 

'48  54  56  48  54  56 

2.  Count  by  6's  to  54  ;  by  9's  to  54. 

3.  Count  by  8's  to  56  ;  by  7's  to  56. 

4.  How  many  days  are  there  in  9  weeks  ? 

5.  What  is  the  area  of  an  oblong  8  in.  by  7  in.  ? 

6.  How  many  pints  are  there  in  6  gallons  ? 

7.  At  $6  a  week,  how  much  will  a  man's  board 
amount  to  in  9  weeks  ? 

8.  What  will  be  the  cost  of  8  tons  of  coal  at  $  7 
a  ton  ? 

9.  A  boy  earns  $9  each  month.     How  much  will 
he  earn  in  6  months  ? 

10.  At  6  for  1  cent,  how  many  marbles  can  a  boy 
buy  for  8  cents  ? 

11.  At  8^  each,  how  much  will  7  melons  cost  ? 

12.  A  grocer  sold  raisins  at  8^  a  pound.  A  girl 
bought  6  lb.,  and  handed  the  grocer  a  half-dollar. 
How  much  change  should  she  receive  ? 

13.  A  girl  bought  7  yards  of  lace  at  8^  a  yard. 
She  handed  the  clerk  60  ^.  How  much  change  should 
she  receive? 

14.  A  boy  earns  $  10  a  month  and  spends  $  3.  How 
much  will  he  save  in  8  months  ? 


100  MULTIPLICATION    AND    DIVISION 


226.    Study  Exercises. 

8             9             8 

6 

6 

7 

X  6         X  6         X  7 

X  8 

X  9 

X  8 

Study  the  above  exercises  until  you  can  give  the 
products  readily. 

Give  the  products  from  right  to  left,  adding  the 
following  numbers  to  each  product :  6,  7,  8,  9. 

227.  Written  Exercises. 

1.  789,879  X  56  lo.  878,787  x  67  '  i9.'  465,646  x  89 

2.  879,789x56  ii.  687,867x67  2o.i  645,465  x  89 

3.  687,978  x  56  12.  758,578  x  67  21.S  536,463  x  89 

4.  968,786  X  56  13.  847,687  x  67  22>i^345,643  x  89 

5.  759,857x56  i4.  876,543x67  23.5'654,321  x  89 

6.  847,498  X  56  15.  345,678  x  67  '  24.^123,456  x  89 

7.  938,739  X  56  le.  235,786  x  67  25./605,640  x  89 

8.  782,728x56  17.  768,547x67  26/546,365x89 
9..  975,985  X  56  la  647,835  x  67  27.^  435,620  x  89 

228.  Written  Exercises. 

1.  Multiply  Exercises  1-9  above  by  34. 

2.  Multiply  Exercises  10-18  above  by  23. 

3.  Multiply  Exercises  19-27  above  by  65. 

229.  Solve: 

a  .       b  c  d  e 

$924.37  $810.35  $735.41  $806.31  $848.12 
-  $235.49  -  $316.58  -  $236.46 -  $216.73 -  $250.27 


DTVISTON  — LESSON  I  -161 

DIVISION  — LESSON   I 

230.    1.    Memorize  the  following  :  '■ 

_8  _^  _8  _6  _6  __7 

6)48         6)54         7)56         8)48         9)54         8)56 

2.  What  part  of  48  is  6  ?     What  part  of  48  is  8  ? 

3.  Eight  is  what  part  of  56  ?  Six  is  what  part 
of  54? 

4.  What  is  the  ratio  of  6  to  54  ?     Of  54  to  6  ? 

5.  How  many  times  is  the  unit  $8  contained  in 
the  quantity  $48? 

6.  If  $48  is  divided  into  8  equal  amounts,  how 
many  dollars  will  there  be  in  each  part  ? 

7.  If  $9  is  the  unit  that  represents  the  cost  of  1 
table  and  $54  is  the  quantity  that  represents  the 
cost  of  the  tables  bought,  how  many  tables  were 
bought  ? 

8.  A  girl  spent  56  days  with  her  aunt.  How  many 
weeks  did  she  spend  with  her  ? 

9.  A  girl  spent  48^  for  lace  that  cost  8j^  a  yard. 
How  many  yards  did  she  buy  ? 

10.  A  box  containing  6  lb.  of  raisins  was  bought 
for  54^.     How  much  did  the  raisins  cost  per  pound? 

11.  Eight  pounds  of  sugar  were  bought  for  48^. 
What  was  the  cost  of  the  sugar  per  pound  ? 

12.  A  boy  spent  56^  in  7  weeks.     What  was  the 
average  amount  spent  each  week  ? 

1st  Rk  Aimtii — 11 


162  MULTIPLICATION   AND   DIVISION 

231.    Study  Exercises. 

6)48       6)54       7)56       8)l8       9)54       8)~56 


6)49 

6)55 

7)57 

8)49 

9)55 

8)57 

6)50 

6)56 

7)58 

8)50 

9)56 

8)58 

6)51 

6)57 

7)59 

8)51 

9)57 

8)59 

6)52 

6)58 

7)60 

8)52 

9)58 

8)60 

6)53 

6)59 

7)61 

8)53 

9)59 

8)61 

Study  the  above  exercises  until  you  can  give  the 
quotients  without  hesitation. 

Give  the  quotients  and  the  remainders,  if  any. 
Review  Exercise  218,  p.  154. 

232.   Written  Exercises. 

Use  the  numbers  above  the  columns  as  divisors : 


6,5,4 

7,6,4 

8,5,3 

9,2,4 

a 

6 

c 

d 

1. 

484,236 

564,942 

484,032 

544,536 

2. 

460,830 

212,835 

449,296 

598,887 

3. 

496,968 

549,927 

493,328 

580,158 

4. 

508,148 

619,766 

451,392 

482,904 

5. 

533,262 

607,936 

373,248 

273,834 

6. 

473,268 

337,974 

203,720 

136,926 

7. 

496,428 

198,695 

267,544 

122,007 

8. 

109,110 

250,026 

133,088 

194,868 

9. 

995,316 

117,936 

195,696 

220,950 

10. 

218,904 

898,352 

923,624 

597,987 

MULTIPLICATION  —  LESSON  J  163 

MULTIPLICATION  — LESSON  J 


233.    1. 

Memorize  the  foi 

lloiving : 

9 

9             8 

7 

8 

9 

x7 
63 

x8          x8 

72          64 

x9 
63 

x9 

72 

x9 
81 

2.  Count  by  8's  to  72 ;  by  9's  to  81. 

3.  Count  by  7's  to  63  ;  by  6's  to  54. 

4.  A  furniture  dealer  sold  8  tables  at  $8  each. 
How  much  did  he  receive  for  them  ? 

5.  What  is  the  area  of  an  oblong  9  inches  long 
and  8  inches  wide  ? 

6.  What  is  the  area  of  a  flower  bed  9  ft.  long  and 
7  ft.  wide  ? 

7.  At  8^  a  box,  how  much  will  8  boxes  of  berries 
cost  ? 

8.  What  is  the  sum  of  nine  9's  ?    Of  eight  g's  ? 

9.  There  are  9  square  feet  in  1  square  yard.     How 
many  square  feet  are  there  in  8  square  yards  ? 

10.  A  girl  bought  9  yards  of  cloth  at  7^  a  yard. 
She  handed  the  clerk  75^.  How  much  change  should 
she  receive  ? 

11.  How  many  square  inches  are  there  in  the  sur- 
face of  a  piece  of  paper  8  in.  wide  and  9  in.  long  ? 

12.  Find  the  cost  of  7  lb.  of  raisins  at  9^  a  pound. 

13.  What  is  the  ratio  of  8  to  72  ?    Of  9  to  72  ? 


164  MULTIPLICAXrON    AND   DIVISION 


234.  Study  Exercises. 

9     9     8 

7 

8 

9 

x7    x8    x8 

x9 

x9 

x9 

Study  the  above  exercises  until  you  can  give  the 
products  without  hesitation. 

Give  the  products  from  right  to  left,  adding  the 
following  to  each  product :  6,  9,  7,  8. 

235.   Written  Exercises. 

Multiply  the  numbers  in  each  column  by  the  follow- 
ing numbers  :  89,  67,  45,  32,  30  : 


1. 

a 

987,798 

b 
543,435 

c 
859,437 

d 
163,530 

2. 

798,978 

454,353 

627,849 

242,607 

3. 

679,896 

345,345 

790,486 

387,652 

4. 

867,897 

534,354 

936,748 

559,608 

5. 

967,898 

435,435 

382,197 

772,002 

6. 

789,679 

253,524 

904,382 

659,003 

7. 

986,789 

425,342 

678,452 

870,004 

8. 

987,698 

530,420 

987,654 

489,603 

9. 

896,789 
836.  Solve: 

315,402 

456,789 

378,960 

a 

6 

c        d 

e 

1. 

85,123 

96,317   94,164   86,543 

74,239 

-47,536  ■ 

-67,429  -57,369  -18,565 

-24,571 

2. 

68,036 

92,228   87,514   83,634 

84,410 

-40,068 

-65,143  -28,436  -34,057 

-47,565 

DIVISION  — LESSON  J  165 

DIVISION  — LESSON  J 

237.    1.    Memorize  the  folloioing : 

_9  9  _^  JT  _8  __9 

7)68  8)72"         8)64  9)63  9)72  9)81 

2.  What  is  \  of  $  63  ?     What  is  i  of  $  72  ? 

3.  At  8^  a  box,  how  many  boxes  of  berries  can  be 
bought  for  64^? 

4.  A  boy  paid  63^  for  9  pounds  of  raisins.  How 
much  per  pound  was  this  ? 

5.  Nine  boys  paid  for  the  lemonade  for  a  class 
picnic.  The  cost  of  the  lemonade  was  81^.  What 
was  each  boy's  share  of  the  expense  ? 

6.  Eight  girls  gave  a  party.  The  expenses 
amounted  to  72^.  What  was  each  girl's  share  of 
the  expenses? 

7.  At  $8  each,  how  many  tables  can  be  bought 
for  $64? 

8.  What  is  the  unit  of  measure  in  Problem  7  ? 

9.  Sixty-three  trees  were  set  out  in  7  rows  with 
the  same  number  of  trees  in  each  row.  How  many 
trees  were  set  in  each  row  ? 

10.  An  oblong  containing  72  sq.  in.  is  9  in.  long. 
How  wide  is  it  ? 

11.  How  many  pounds  of  candy  at  9^  a  pound  can 
be  bought  for  72^? 


166  MULTIPLICATION   AND  DIVISION 

238.    Study  Exercises. 


7)63 

8)72 

8)64 

9)63 

9)72 

9)81 

7)64 

8)73 

8)65 

9)64 

9)73 

9)82 

7)65 

8)74 

8)66 

9)65 

9)74 

9)l53 

7)66  8)75         8)67  9)66  9)75         9)84 

Study  the  above  until  you  can  give  the  quotients 
without  hesitation. 

Give  the  quotients  and  remainders,  if  any. 

Give  the  quotients  with  the  remainders  expressed 
as  fractions. 

Increase  the  dividends  in  each  of  the  above  columns 
until  each  divisor  is  contained  in  the  dividend  10  times. 

Review  Exercises  218  and  231. 

239.   Written  Exercises. 

Use  the  numbers  above  the  columns  as  divisors : 


a 

b 

C 

d 

7,  5 

8,  6,  4 

3,  9,  2 

4,  5 

1. 

635,649 

726,456 

978,560 

322,789 

2. 

213,542 

324,048 

384,271 

263,867 

3. 

714,644 

241,632 

567,894 

545,982 

4. 

286,573 

738,768 

306,752 

756,789 

5. 

698,572 

516,976 

983,084 

295,837 

6. 

678,573 

206,344 

542,301 

106,967 

7. 

599,186 

767,592 

894,517 

408,432 

8. 

608,033 

769,896 

675,214 

702,002 

ORAL  PROBLEMS  167 

240.    Oral  Problems.* 

1.  If  4  chairs  cost  $12,  how  many  chairs  can  be 
bought  for  $21? 

Model  for  oral  recitation :  If  4  chairs  cost  $  12, 
1  chair  will  cost  ^  of  $12,  or  $3.  If  1  chair  costs 
$3,  as  many  chairs  can  be  bought  for  $21  as  there 
are  $3  in  $21,  or  7.    7  chairs  can  be  bought  for  $21. 

Model  for  written  recitation : 

$3,  cost  of  1  chair. 
12,  cost  of  4  chairs. 

7  chairs  for  ^21. 


cost  of  1  chair,  $3)$  21 

2.  If  3  tables  cost  $18,  how  many  tables  can  be 
bought  for  $30? 

3.  If  2  tons  of  coal  cost  $14,  how  many  tons  can 
be  bought  for  $28? 

4.  How  many  yards  of  ribbon  can  be  bought  for 
20^,  if  3  yd.  cost  12^? 

5.  How  many  tablets  can  be  bought  for  40^,  if 
3  tablets  cost  24^? 

6.  At  the  rate  of  3  boxes  for  18^,  how  many  boxes 
of  berries  can  be  bought  for  36^  ? 

7.  How  many  pencils  can  be  bought  for  24^,  if 
3  pencils  cost  9^? 

8.  If  Alice  uses  6  lemons  to  make  2   pies,  how 
many  pies  will  a  dozen  lemons  make  ? 

*  See  note,  p.  115. 


168  MULTIPLICATION  AND  DIVISION 

241.    Oral  Problems. 

Give  the  analysis  for  each : 

1.  How  much  will  4  trunks  cost  at  $  7  each  ? 

2.  A  boy  bought  36  ^  worth  of  sugar  at  G  ^  a  pound. 
How  many  pounds  did  he  buy  ? 

3.  A  grocer  sold  6  qt.  of  berries  at  8^  a  quart. 
How  much  did  he  receive  for  them  ? 

4.  If  6  pairs  of  shoes  cost  $  18,  what  is  the  cost 
of  4  pairs  of  shoes  ? 

5.  How  many  desks  can  be  bought  for  |  32,  if  6 
desks  cost  $  24  ? 

6.  How  far  will  a  boy  ride  in  8  hours,  if  he  travels 
at  the  rate  of  12  miles  in  3  hours  ? 

7.  If  5  apples  cost  10^,  how  much  will  8  apples 
cost  at  the  same  rate  ? 

8.  If  4  bunches  of  firecrackers  cost  20^,  how  much 
will  6  bunches  cost  at  the  same  rate  ? 

9.  How  many  sacks  of  potatoes  at  $  2  a  sack  will 
pay  for  4  tons  of  coal  at  $  6  a  ton  ? 

10.  If  it  takes  a  boy  12  minutes  to  ride  f  of  a  mile, 
how  long,  at  the  same  rate,  will  it  take  him  to  ride  a 
mile  ? 

11.  If  it  takes  2  men  1  day  to  build  a  fence,  in 
what  time  can  1  man  build  it  ? 

12.  If  an  acre  of  land  is  worth  $  45,  how  much 
is  I"  of  an  acre  worth  at  the  same  rate  ? 

13.  A  grocer  sold  |-  of  a  box  of  apples  for  40^. 
How  much  was  the  box  worth  at  the  same  rate  ? 


WRITTEX   PROBLEMS  169 

213.    Written  Problems. 

1.  A  farmer  had  360  acres  of  land.  He  sold  f  of 
it  at  $  60  an  acre.     How  much  did  he  receive  for  it  ? 

2.  A  man  sold  f  of  his  farm  for  $4200.  What, 
at  the  same  rate,  was  the  value  of  the  farm  ? 

3.  A  boy  had  $36  in  a  bank.  He  drew  out  f  of 
it  to  pay  for  a  bicj^cle.  How  much  did  the  bicycle 
cost  him  ? 

4.  In  an  orchard  containing  480  trees  |  of  the 
trees  are  orange  trees  and  the  remainder  are  lemon 
trees.  Find  how  many  of  each  kind  there  are  in  the 
oTchard. 

5.  A  farmer  sold  3  cows  for  $  45  each  and  2  horses 
for  $130  each.  He  deposited  in  a  bank  f  of  the 
money  received.     Find  the  amount  of  his  deposit. 

6.  A  man  bought  8  horses  at  an  average  cost  of 
$79.  He  sold  them  all  for  $760.  How  much  did 
he  make  on  them  ?  Find  the  average  amount  made 
on  each  horse. 

7.  A  man  bought  9  cows  for  $225  and  sold  them 
for  $315.  How  much  did  he  make  on  them?  Find 
the  average  amount  made  on  each  cow. 

8.  A  farmer  sold  360  sacks  of  potatoes,  which  was 
I  of  his  entire  crop,  at.  $2  a  sack.  What  was  the 
value  of  his  entire  crop  at  the  same  rate  ? 

9.  How  many  weeks  must  a  boy  work  at  $3  a 
week  to  pay  for  a  suit  of  clothes  that  costs  $  12  ? 


170 


MULTIPLICATION   AND  DIVISION 


243.    Drill  Exercises. 

Give  quotients  with  the  remainders  expressed  as 
fractions : 


1. 

¥>  ¥>  ¥• 

7. 

¥.  ¥>  ¥- 

13. 

¥>  ¥,  ¥- 

2. 

¥.  ¥,  ¥• 

a 

¥,  ¥.  ¥• 

14. 

¥,  ¥.  ¥- 

3. 

¥->  ¥>  ¥• 

9. 

¥>  ¥-  ¥• 

15. 

¥-.¥,¥• 

4. 

¥,  ¥.  ¥• 

lo. 

¥>  ¥'  ¥• 

16. 

\S  ¥,  ¥• 

5. 

¥>  ¥,  ¥• 

11. 

¥>  ¥,  ¥- 

17. 

¥.  ¥,  ¥• 

6. 

¥>  -¥>  ¥• 

12. 

¥v¥.  -V- 

18. 

¥,  ¥.  ¥• 

244.    Oral  Exercises. 

Add  each  column  as  indicated  in  Step  C,  p.  49. 


a 

6 

c 

d 

e 

/ 

S' 

h 

i 

J 

fc 

; 

m 

6 

8 

3 

6 

8 

3 

8 

.  2 

8 

3 

4 

4 

3 

3 

4 

2 

2 

4 

6 

4 

3 

7 

9 

2 

8 

9 

2 

5 

3 

2 

5 

8 

9 

4 

6 

7 

3 

5 

7 

6 

8 

9 

C 

8 

3 

5 

5 

5 

2 

7 

4 

5 

3 

6 

2 

2 

4 

6 

6 

6 

4 

3 

2 

2 

2 

2 

4 

3 

2 

5 

4 

8 

7 

3 

6 

5 

3 

4 

6 

5 

7 

7 

3 

3 

3 

8 

2 

4 

3 

8 

8 

8 

8 

9 

8 

7 

2 

7 

9 

1 

7 

4 

9 

3 

Two  addends  whose  sum   is  ten  or  less  may  be 
taken  as  one  number. 

Exercise  a  above  may  be  added :  14,  19,  27,  36. 

It  may  also  be  added  :   16,  25,  30,  36. 

Add  the  above  exercises  in  a  similar  manner. 


WRITTEN  PROBLEMS  171 

245.    Written  Problems. 

1.  A  man  bought  9  horses  for  $  945  and  sold 
them  for  $125  each.  Find  the  amomit  of  his  gain 
or  loss.  What  was  the  average  gain  or  loss  on  each 
horse  ? 

2.  A  grocer  bought  6  barrels  of  apples  at  $3.25 
per  barrel  and  sold  the*m  all  for  $25.  Find  the 
amount  of  his  gain  or  loss. 

3.  A  farm  of  189  acres  was  bought  for  $65  an 
acre  and  sold  for  $80  an  acre.  Find  the  gain  per 
acre.     Find  the  gain  on  the  entire  farm. 

4.  A  hardware  merchant  sold  stoves  at  $9  each. 
During  the  year  he  sold  $405  worth  of  stoves.  How 
many  stoves  did  he  sell  ?  His  profit  was  $  2  on  each 
stove  sold.  How. much  did  he  make  on  the  sale  of 
stoves  during  the  year  ? 

5.  A  man  sold  |-  of  his  farm  for  $3600.  What 
part  of  his  farm  was  left  ?  At  the  same  rate,  what 
was  the  vahie  of  the  part  that  was  left?  What  was 
the  value  of  the  entire  farm  ? 

6.  A  harness  cost  $30.  This  was  f  of  the  cost  of 
the  buggy.  Find  the  cost  of  the  1  i^y.  The  har- 
ness and  buggy  together  cost  ^  of  ^Le  cost  of  the 
horse.  Find  the  cost  of  the  horse.  Find  the  cost  of 
all  three. 

7.  A  man  had  288  miles  to  travel.  He  rode  f  of 
the  distance  on  his  wheel  and  the  remainder  on  a 
train.     How  far  did  he  ride  on  each  ? 


172  MULTIPLICATION    AND   DIVISION 


MULTIPLICATION 

-LESSON  K 

246. 

Oral  Exercises. 

7 

75             64 

70          354 

100 

xlO 

xlO          xlO 

xlO         xlO 

X  10 

70 

750           640 

700        3540 

1000 

1.  Compare  the  products  in  the  above  exercises 
with  the  multiplicands.     How  do  they  diit'er  ? 

2.  Can  you  give  a  short  method  of  multiplying  a 
number  by  1 0  ? 

3.  Multiply  each  of  the  following  numbers  by  10 : 
34,  45,  8,  90,  11,  12,  524,  670,  200. 

4.  Can  you  give  a  short  method  of  multiplying  a 
number  by  1 00  ? 

5.  Multiply  each  of  the  numbers  in  Problem  3  Ijy 
100. 

6.  Each   of   the  following  numbers   is  10  times 
what  number :  60,  80,  90,  950,  100,  300,  760,  7600  ? 

7.  Each  of  the  following  numbers  is  100  times 
what  number :  600,  100,  5000,  3700,  9500,  85,000  ? 

8.  Divide  each  of  the  following 'numbers' by  10: 
80,  110,  210,  340,  450,  6050. 

9.  Give  a  short  method  of  dividing  a  number  that 
ends  in  zero  by  10. 

10.  A  number  that  does  not  end  in  zero  may  be 
divided  by  10  thus:  85-?- 10  is  8.5.  A  point  called  a 
decimal  point  is  placed  before  the  right-hand  figure 
of  the  number.  This  divides  it  by  10.  The  answer 
is  read,  e'uiht  (Uid five  tenths.     It  is  the  same  as  S^^''^. 


MULTIPLICATION  — LESSON   L  173 


MULTIPLICATION - 

-LESSON  L 

247. 

Oral  Exercises. 

3 

4               5 

6               11 

11 

xll 

xll          xll 

xll           x4 

x6 

33 

44             55 

66            44 

66 

Study  these  exercises  to  find  a  short  way  of  mul- 
tiplying any  number  from  1  to  9  by  11.  Multiply 
by  11:  8,  7,4,  3,  1,2,  9,  6,  5. 

Multiply  11  by  each  of  the  above  numbers. 

MULTIPLICATION  — LESSON  M 
248.    Oral  Exercises. 

1.  The  number  12  is  used  in  many  of  our  measure- 
ments. There  are  12  inches  in  1  foot ;  there  are 
12  months  in  1  year ;  there  are  12  things  in  1  dozen ; 
and  the  clock  face  is  divided  into  12  parts.  Twelve 
dozen  things  are  sometimes  put  together  and  called  a 
gross.     Is  there  a  gross  of  crayon  in  a  full  box  ? 

2.  Memorize : 


12 

12 

12 

12 

12 

x3 

x4 

x5' 

x6 

x7 

36    • 

48 

60 

7^ 

84 

12 

12 

12 

12 

12 

x8 

x9 

xlO 

xll 

xl2 

96 

108 

120 

132 

144 

3.  How  many  oranges  are  5  doz.  oranges  ? 

4.  How  many  months  are  9  years?  12  years? 


174  MULTIPLICATION   AND   DIVISION 

249.   Table  of  Products  and  Quotients. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 
15 

16 
20 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72' 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

LONG  MEASURE* 

250.  Distance  is  measured  in  inches,  feet,  yards, 
rods,  and  miles.  The  yard  is  the  standard  unit  of 
length.     The  other  units  are  derived  from  it. 

251.  Menwjize : 

12  inches  (in.)  =  1  foot  (ft.) 

3  feet  =  1  yard  (yd.) 

5 1  yards,  or  16 J  feet=  1  rod  (rd.) 
320  rods  =  1  mile  (mi.) 

Imile  =1760  yd.  =  5280  ft. 

*  Review  Exercise  140,  p.  114. 


SQUARE   MEASURE  175 

SQUARE  MEASURE 

252.  1.  Using  a  yardstick,  draAV  on  the  blackboard 
a  square  whose  side  is  one  yard.  This  is  a  square 
yard. 

2.  Using  a  foot  rule,  draw  a  square  whose  side  is 
one  foot.     What  is  this  square  called  ? 

3.  Divide  the  square  yard  into  square  feet.  How 
many  square  feet  are  there  in  one  square  yard  ? 

4.  Draw  a  square  inch.  Divide  a  square  foot  into 
square  inches.  How  many  square  inches  are  there 
in  one  square  foot  ?     12x12  =  — . 

5.  Measure  on  the  ground  a  square  whose  side  is 
one  rod.  Drive  a  stake  at  each  corner  of  it.  This 
is  called  a  — . 

6.  It  takes  160  square  rods  to  make  one  acre.  A 
piece  of  land  16  rd.  long  and  10  rd.  wide  is  one  acre. 
A  piece  of  land  20  rd.  long  and  —  rd.  wide  is  one  acre.''^ 

7.  A  piece  of  land  one  mile  square  contains  640 
acres.     This  is  called  a  section. 

253.  Memorize  : 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  square  feet  =  1  square  yard  (sq.  yd.) 

30^  square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

*If  feasible,  have  the  children  measure  off  one  acre  on  the  school 
grounds  or  in  an  adjoining  field. 


176  Mri/riPLICATTON    AND    DIVISION 

254.  Written  Problems. 

1.  Find  the  number  of  square  feet  in  a  walk  50  ft. 
long  and  4  ft.  wide. 

2.  How  much  will  it  co^t  at  $  .12  a  square  foot 
to  lay  a  cement  walk  50  ft.  long  and  5. ft.  wide  ? 

3.  Find  the  area  of  a  walk  60  ft.  long  and  5  ft. 
wide. 

4.  How  many  square  inches  are  there  in  o  sq.  ft.? 
In  31  sq.  ft.  ? 

5.  How  many  square  inches  are  there  in  3  sq.  ft.  and 
32sq.  in.? 

6.  Find  the  number  of  square  feet  there  are  on 
the  floor  of  your  schoolroom. 

7.  Find  the  number  of  square   rods  there  are  in 
your  school  yard. 

8.  Find  the  number  of  square  yards  of  blackboard 
there  are  in  your  schoolroom. 

9.  How  much  did  the  blackboard  in  your  school- 
room cost  at  18^  a  square  foot? 

255.  In  your  drawings,  let  1  in.  represent  2  ft. 
Draw : 

1.  A  square  that  will  contain  16  sq.  ft. 

2.  A  rectangle  that  will  contain  16  sq.  ft. 

3.  A  square  that  will  contain  36  sq.  ft. 

4.  A  rectangle  that  will  contain  36  sq.  ft. 

5.  Find  the  perimeter  of  each  of  your  figures. 


BILLS   AND   ACCOUNTS 


177 


BILLS  AND  ACCOUNTS 

256.    1.    Study  the  bill  given  in  Sec.  78,  p.  82. 

2.  A  bill  must  always  show  the  date  of  the  trans- 
action. What  is  the  date  of  the  transaction  re- 
ferred to  in  the  bill  in  Sec.  78  ? 

3.  The  debtor  is  the  party  who  buys  the  goods. 
Who  is  the  debtor  in  the  bill  referred  to  above  ? 

4.  The  creditor  is  the  party  who  .sells  the  goods. 
Who  is  the  creditor  in  the  bill  referred  to  above  ? 

5.  An  item  is  a  separate  debit  or  credit  made  in 
a  bill.  How  many  items  are  there  in  the  bill  referred 
to  above  ? 

6.  How  many  items  are  there  in  the  bill  in  Sec. 
257? 

7.  Name  the  debtor  and  the  creditor  in  the  bill  in 
Sec.  257. 


257. 

Henry  Love, 


A   RECEIPTED    BILL 

Oakland,  Cal.,  June  30,  1904. 

Bought  o/ James  Roland. 


5  lb.  sugar 

2  caus  tomatoes 

2  lb.  coffee 


.05 
.10 
.40 


Received  Payment, 
James  Roland. 


.25 
.20 

.80 


2^ 


Make  similar  bills,  using  regular  bill  paper. 

1st  Bk  Aritii — 12 


178  MULTIPLICATION   AND  DIVISION 

LONG  DIVISION 

258.  When  all  the  steps  in  division  are  written,  the 
process  is  called  long  division. 

Divide  173  by  3. 

Model:         57f  8  is  contained  in  17  five  times. 

3)173        Write  5  in  the  quotient  above  the 
15  7,  as  in  short  division.     Multiply  3 

23       by  5  and  write  the  product  under 
21        17,  and  subtract.     The  remainder  is 
2        2.     Bring  down  the  3  of  the  divi- 
dend and  find  how  many  times  3  is 
contained  in  23.     This  is  7  times.     Multiply  3  by  7  and 
write  the  product  under  23,  and  subtract.     Treat  the  re- 
mainder as  in  short  division. 

CASE  ONE 

259.  When  the  second  figure*  of  the  divisor  is  the 
same  as  or  less  than  the  first  figure,  as  in  44,  63,  978, 
658,  etc. 

1.   Divide  2292  by  43. 

First,  find  how  many  places  at  the  left  of  2292  it  will 
take  to  contain  43  at  least  one  time.  It  will  take  three 
places.     The  first  figure  of  the  quotient  will  be  in  tens' 

,,  ^ON^nr^r>  ^i^P  1-    4  is  contained  in  22 

Model:  43)2292         n        f.  •  ,     .. 

ey^r  five    times    with    2    remainder. 

~TT^         This   remainder,  with    the  next 

-190         figure    of    the    dividend,    is   the 

^r^         dividend  of  3,  the  second  figure 

of  the  divisor.     The  dividend  of 

*  In  63  regard  6  as  the  first  fin:ure  and  3  as  the  second  figure. 


LONG  DIVISION  179 

3  is  29.  Is  3  contained  in  29  as  many  as  5  times  ?  If 
it  is,  5  is  the  trial  quotient  figure.  3  is  contained  in  29 
as  many  as  5  times.     Write  5  in  the  quotient  above  9. 

Step  2.  Multiply  43  by  5,  and  write  the  product 
under  229. 

StepZ.     Subtract  215  from  229.     The  remainder  is  14. 

As  this  remainder  is  less  than  the  divisor,  the  trial 
quotient  figure  is  the  true  quotient  figure. 

Step  4.  Bring  down  the  next  figure  of  the  dividend. 
The  new  dividend  is  142.  Repeat  Step  1.  4  is  contained 
in  14  three  times  with  2  remainder.  As  3  is  contained 
in  22  as  many  as  3  times,  3  is  the  trial  quotient  figure. 
Write  3  in  the  quotient. 

Repeat  Step  2.  Multiply  43  by  3  and  write  the  product 
under  142. 

Repeat  Step  3.  Subtract  129  from  142.  The  remainder 
is  13.     Treat  the  remainder  as  in  short  division. 

2.   Divide  1806  by  43  ;'V877  by  96  ;'\877  by  82. 

3.^'  Divide  2806  by  65  ;' 5927  by  97  ;  5927  by  51. 

4.^^'Divide  16,108  by  72;  48,191  by  85;  "45,960 
by  71. 

5.   Divide  2115  by  43. 

First,  find  how  many  places  at  the  left  of  2115  it  will 
take  to  contain  43  at  least  one  time. 

Model  :  49^3         Step  1.     4  is  contained  in  21 

43)2115  five  times  with  1  remainder.     As 

172  3  is  not  contained  in  11  as  many 

395  as  5  times,  write  4  as  the  trial 

387  quotient  figure,  and  continue  as 

8  in  the  preceding  exercise.     - 


180  MULTIPLICATION  AND  DIVISION 

6.  Divide  6748  by  76  ;  8482  by  54 ;  7667  by  99. 

7.  Divide  4084  by  72;   2094  by  93  ;   6456  by  77. 

a  Divide  1243  by  22;  4527  by  33;  5468  by  76. 

9.  Divide  9247  by  95. 

q  9  is  contained  in  92  ten  times. 

Model:    95)¥247      Use  9  as  the  trial  quotient  figure. 
Complete  the  division. 

10.  Divide  7492  by  77 ;  6230  by  65 ;  9348  by  98. 

11.  Divide  6258  by  66  ;  5213  by  55  ;  8570  by  87. 

12.  Divide  5234  by  54 ;  7485  by  76 ;  8775  by  88. 

260.  When  the  second  figure  of  the  divisor  is  the  same 
as  or  less  than  the  first  figure^  the  trial  quotient  figure  may 
he  found  as  follows : 

1.  If  the  second  figure  of  the  divisor  is  contained  in  its 
dividend  *  as  many  times  as  the  first  figure  is  contained  in 
its  dividend^  use  this  quotient  figure  as  the  trial  quotient 
figure, 

2.  If  the  second  figure  of  the  divisor  is  not  contained  in 
its  dividend  as  many  times  as  the  first  figure  is  contained  in 
its  dividend^  use  as  a  trial  quotient  figure  one  less  than  the 
quotient  figure  obtained  by  dividing  by  the  first  figure  of  the 
divisor.      This  unll  he  found  to  he  the  true  quotient  figure. 

3.  //  the  first  figure  of  the  divisor  is  contained  in  its 
dividend  10  times^  use  9  as  the  trial  quotient  figure.^ 

*  The  number  formed  by  annexing  the  next  figure  of  the  dividend 
to  tlie  remainder  left  after  dividing  by  the  first  figure  of  the  divisor  is  the 
dividend  of  the  second  figure  of  the  divisor. 

t  The  pupils  should  become  perfectly  familiar  with  these  facts  through 
illustration. 


LONG   DIVISION  181 

With  divisors  of  two  places,  the  trial  quotient  figure 
obtained  as  indicated  in  1  and  3  will  be  found  to  be  the 
true  quotient  figure.  Often  with  divisors  of  more  than 
two  places,  the  trial  quotient  figure  will  be  found  to  be 
one  more  than  the  true  quotient  figure. 

261.  Written  Exercises. 

Examine  the  divisors  used  in  these  exercises. 
Divide  : 

1.  51,913  by  54.  12.  86,734  by  85. 

2.  65,760  by  87.  13.  67,863  by  75. 

3.  51,596  by  92.  i4.  54,326  by  64. 

4.  61,640  by  76.  15.  92,147  by  53. 

5.  51,594  by  96.  le.  92,147  by  534. 
6..*83,756by  43.  17.  54,326  by  647. 
7.J.  98,765  by  21.  la  94,245  by  736. 
8.i  45,637  by  86.  19.  67,863  by  754. 
9.  ^^94,245  by  73.  20.  67,321  by  959. 

10.5.87,653  by  54.  21.    51,504  by  967. 

11.    67,321  by  9^.  22.    45,637  by  865. 

262.  Written  Exercises. 

Examine  each  divisor  before  using  it. 

Divide  each  of  the  following  by  87,  94,  63,  72,  55, 
22,31,44,33,652^773,940: 

1.  93,456.       3.    54,943.       5.    19,831.       7.   40,572. 

2.  67,342.       4.    86,425.       6.    24,753.       s.   98,345. 


182  MULTIPLICATION   AND  DIVISION 

363.   Written  Problems. 

1.  There  are  52  weeks  in  one  year.  How  many 
years  are  there  in  468  weeks  ? 

2.  A  man  bought  a  carload  of  cattle  at  $32  each. 
He  paid  $800  for  the  carload.  Find  the  number  of 
cattle  in  the  car. 

3.  A  train  travels  at  an  average  speed  of  42  mi. 
an  hour.  How  many  hours  will  it  take  it  to  travel 
1000  mi.  ? 

4.  A  hardware  merchant  paid  $21  each  for  stoves. 
The  amount  of  his  bill  was  $315.  How  many  stoves 
did  he  buy  ? 

5v  A  dealer  bought  a  carload  of  horses  at  $95 
each.     He  paid  $2565  for  them.     How  many  horses 

did  he  buy  ? ___..._._.«.«-«««-.i.--------— -—- 


6.  A  merchant  paid  $3570  for  some  carriages  at 
$210  each.     How  many  carriages  did  he  buy? 

7.  At  $  65  a  month  J  in  how  many  months  will  a 
clerk  earn  $  975  ? 

8.  At  55^  a  yard,  how  many  yards  of  cloth  can 
be  bought  for  $  8.25  ?     (Change  to  cents.) 

9.  At  75^  each,  how  many  tickets  must  be  sold  to 
amount  to  $15  ? 

10.  There  were  42  children  at  a  school  picnic. 
The  expenses  of  the  picnic  were  $  10.50.  Find  each 
one's  share  of  the  expenses. 


LONG  DIVISION  183      - 

CASE  TWO 

264.  When  the  second  figure  of  the  divisor  is 
greater  than  the  first  figure,  as  in  37,  28,  287, 
596,  etc. 

Divide  1734  by  47. 

o  When  the  divisor  is  47,  48,  or  49, 

M  .  dTvPT^J    ^^®  ^  ^^  ^^®  divisor  to  find  the  trial 

quotient  figure.  5  is  contained  in 
17  three  times.  Use  3  as  the  trial  quotient  figure.  Com- 
plete the  division. 

When  the  second  figure  of  the  divisor  is  7,  8,  or  9,  the 
number  to  be  used  as  a  divisor  to  find  the  trial  quotient 
figure  is  determined  as  follows : 

When  the  divisor  is  47,  48,  49,  475,  etc.^  use  5  as  a 
divisor.  When  the  divisor  is  57,  58,  59,  584,  etc.^  use  6 
as  a  divisor. 

The  trial  quotient  figure  thus  obtained  is  sometimes 
one  more  or  one  less  than  the  true  quotient  figure. 

265.  Written  Exercises. 

Before  dividing,  state  what  divisor  will  be  used  to 
find  the  trial  quotient  figure  in  each  of  the  following. 
Divide : 

1.  54,321  by  19.  o^  i.  71,067  by  38.    13.  43,907  by  275. 

2.  54,796  by  38.  5  8.  30,402  by  49.    i4.  69,087  by  594. 

3.  12,345  by  69.  t  9.  42,796  by  59.    15.  87,906  by  178. 

4.  43,697  by  28.>  10.  74,908  by  67.    16.  38,690  by  576. 

5.  43,345  by  18.   11.  43,768  by  57.    17.  43,234  by  374. 
6.^54,678  by  57.   12.  67,098  by  19.    is.  41,908  by  490. 


184  MULTIPLICATION   AND   DIVISION 

266.  Divide  1085  by  25. 

^  2  is  contained  in  10  five  times. 

MonFT  •     2^  yTTTF^      ^^^  ^^®  ^^^^  than  this  quotient  as  a 
trial  quotient  figure.     4  is  the  trial 
quotient  figure.     Complete  the  division. 

When  23-26,  34-36,  45,  46,  and  56  are  used  as  divisors^ 
or  as  the  first  two  figures  of  divisors,  use  as  a  trial  quotient 
figure  one  less  than  the  quotient  obtained  hg  dividing  the 
first  figure^  or  figures^  of  the  dividend  by  the  first  figure  of 
the  divisor. 

The  trial  quotient  figure  thus  obtained  is  sometimes 
one  more  or  one  less  than  the  true  quotient  figure. 

267.  Written  Exercises. 

Divide  each  by  25, '35,  46,  56,  243,  462,  and  358: 

1.  9875.  7.  9356.  13.  6532.  19.  9530. 

2.  2675.  8.  3640.  i4.  3256.  20.  4780. 

3.  3650.  9.  1884.  15.  7890.  21.  8020. 

4.  4563.  10.  3657.  is.  5231.  22.  6510. 

5.  1895.  11.  7627.  17.  6423.  23.  1560. 

6.  2750.  12.  1840.  18.  5768.  24.  7090. 

268.  Written  Exercises. 

Divide  each  by  65,  49,  36,  245,  and  684  : 

1.  547,659.  5.    634,237.  9.  120,500. 

2.  134,652.  6.   845,178.  10.  887,945. 

3.  347,865.  7.    342,156.  11.  674,109. 

4.  937,311.  a.   240,100.  12.  832,674. 


LONG  DIVISION  185 

269.  When  such  numbers  as  13,  14,  15,  16,  134, 
149,  157,  etc.,  are  used  as  divisors,  the  trial  quotient 
figure  can  not  be  determined  by  any  definite  rule ; 
but  by  the  following  method  a  trial  quotient  figure 
may  be  found  that  will  seldom  vary  more  than  one 
from  the  true  quotient  figure. 

Divide  1067  by  14. 

_  Use  2  as  a  divisor.     2  is  con- 

^  .c^TTTTTs^       tained  in  10  five  times.     Add  2  to 
Model:  14)10b7      ^,  .         ^-     ^.       ^  .  i       x-     .  x^ 
^  this  quotient  lor  a  trial  quotient  ng- 

ure.    7  is  the  trial  quotient  figure.    Complete  the  division. 

When  13,  14,  15,  16, 138,  etc.^  are  used  as  divisors,  divide 
the  first  figure,  or  the  first  two  figures,  of  the  dividend  hy  2, 
a7id  add  2  to  the  quotient  thus  obtained  for  a  trial  quotient 
figure,  unless  the  quotient  figure  can  he  determined  readily 
hy  inspection. 

270.  Written  Exercises. 

Divide : 

1.  95,478  by  15.  6.  13,468  by  16.  ii.  84,675  by  138. 

2.  87,345  by  14.  7.  23,410  by  15.  12.  12,432  by  146. 

3.  11,745  by  16.  8.  46,098  by  13.  13.  90,543  by  164. 

4.  10,000  by  13.  9.  10,710  by  14.  i4.  14,500  by  159. 

5.  20,348  by  14.  10.  12,000  by  15.  15.  15,234  by  163. 

271.  Written  Exercises. 

Divide  each  by  95,  79,  36,  16,  425,  386,  and  145  : 

1.  548,674.  3.  240,575.  5.  450,100. 

2.  427,658.  4.  318,925.  6.  987,689. 


186  MULTIPLICATION  AND  DIVISION 

272.    Written  Exercises. 
Divide : 

1.  25,678  by  38.  6.  23,670  by  15.  hi.  37,896  by  376. 

2.  17,408  by  19.  7.  10,682  by  14.  '12.  51,678  by  645. 

3.  62,389  by  83.  a  25,678  by  38.    13. -10,682  by  210. 

4.  37,896  by  52.  9.  17,408  by  27.    i4.  12,367  by  144. 

5.  51,678  by  57.  10.  62,389  by  88.  j  15.  98,765  by  990. 

16.  There  are  5280  ft.  in  oiie  mile.  Reduce 
5,786,968  ft.  to  miles. 

17.  There  are  365  da.  in  one  year.  Reduce 
^63,475  da.  to  years. 

18.  At  15^  a  gallon,  how  many  gallons  of  oil  can 
be  bought  for  $4.50? 

19.  A  bushel  of  wheat  weighs  60  lb.  A  cental  is 
100  lb.  How  many  bushels  are  there  in  18  centals 
of  wheat  ? 

20.  A  barrel  of  flour  contains  196  lb.  How  many 
barrels  can  be  filled  from  6272  lb.  of  flour  ? 

21.  At  $  75  an  acre,  how  many  acres  of  land  can 
be  bought  for  $  3000  ? 

22.  If  a  man  saves  $  45  a  month,  in  how  many 
montlis  will  he  save  enough  to  buy  a  farm  worth 
$3150? 

23.  There  are  144  sq.  in.  in  one  square  foot.  Re- 
duce 11,520  sq.  in.  to  square  feet. 

24.  There  are  320  rd.  in  one  mile.  Reduce  1920  rd. 
to  miles. 


WRITTEN   PROBLEMS  187 

273.    Written  Problems. 

1.  A  farmer  planted  an  orchard  of  240  trees. 
There  were  24  trees  in  each  row.  How  many  rows 
were  there  in  the  orchard  ? 

2.  Find  the  cost  of  26  cans  of  oil  at  $1.10  a  can. 

3.  A  trader  paid  22  ^  a  dozen  for  eggs.  He  bought 
$8.80  worth.     How  many  dozen  did  he  buy? 

4.  A  car  made  8  trips  in  a  day.  On  an  average 
the  car  carried  85  people  each  trip.  The  fare  was  5)^. 
What  were  the  total  receipts  for  the  day  ? 

5.  If  school  is  in  session  5  hours  a  day  for  200 
days  in  the  year,  how  many  hours  of  school  are  there 
in  a  year  ? 

6.  A  boy's  salary  is  .$20  a  month.  He  has  been 
paid  $120.     How  many  months  has  he  worked? 

7.  A  boy  puts  $  8  in  a  bank  each  month  he  works. 
He  has  $176  in  the  bank.  How  many  months  has 
he  been  working  ? 

8.  '  A  farmer  sold  his  hay  at  $  9  a  ton.  He  received 
for  his  crop  $  1 800.    How  many  tons  of  hay  did  he  sell  ? 

9.  ^  A  farmer  sold  his  wheat  at  $.85  a  bushel.  He  sold 
78  bushels.    How  much  did  he  receive  for  his  wheat  ? 

10.^- A  merchant  buys  flour  at  $5  a  barrel  and  sells 
it  at  $6.  How  much  does  he  gain  on  each  barrel? 
$  1  is  what  part  of  $  5  ? 

11.  A  boy  bought  30  papers  at  the  rate  of  2  for  5^. 
He  sold  25  of  them  at  5^  apiece.  The  other  5  he 
gave  away.     Did  he  gain  or  lose,  and  how  much  ? 


CHAPTER   V 

FRACTIONS   AND   DECIMALS 

COMPOUND  NUMBERS  AND  REVIEWS 

ADDITION  OF  FRACTIONS 

274.  Written  Exercises. 

One  boy  and  1  boy  and  1  boy  are  —  boys. 

One  half  and  one  half  and  one  half  are  —  halves. 

-|  are  equal  to  —  ^. 

Add  :  4^  Add  the  fractions  first :  |  +  J  + 1  are  f  5 
21  which  are  equal  to  1^.  Write  the  ^  under 
2-|  the  colunm  of  fractions  and  add  the  1 
9 J    to  the  column  of  whole  numbers. 

275.  Add  the  following : 


a 

6 

c 

d 

e 

/ 

9 

h 

/ 

H 

8 

9 

H 

7 

74 

7 

H 

6i 

^ 

5 

8 

H 

H 

6 

51 

6 

51 

H 

H 

2* 

% 

H 

9 

6| 

2 

8+ 

9 

ii 

H 

7i 

8 

81 

9 

n 

n 

276.*  1.  Find  by  addition  the  cost  of  5  yards  of 
cloth  at  41^  a  yard. 

2.  How  far  is  it  around  an  oblong  that  is  8^  in. 
long  and  5{>  in.  wide  ?     Draw  the  oblong. 

*  Dictate  additional  problems  of  a  similar  character. 
188 


StBTRACTlOK   OF  FRACTIONS 


189 


277.   Written  Exercises. 
Review  Exercise  90,  p.  88. 


a 

6 

c 

d 

e 

f 

9 

Ji 

H 

H 

H 

H 

4f 

64 

^ 

7f 

54 

5* 

5-1 

6 

n 

54 

H 

94 

64 

n 

6 

7 

5| 

5 

H 

84 

H 

H 

7f 

84 

^ 

6 

H 

74 

^ 

81 

8 

94 

4f 

^ 

5f 

84 

2 

H 

9 

6 

H 

8 

6 

74 

1.  Find  by  addition  ^tlie  distance  around  a  room 
that  is  4|  yd.  long  and  3^  yd.  wide. 

2.  Find  by  addition  tlie  cost  of  6  yd.  of  cloth  at 
5|^  a  yard. 

SUBTRACTION  OF  FRACTIONS 

278.   Written  Exercises. 


*    1.    Subtract  21  from  41.        2.    Subtract  2  from  5J. 

Model  : 

Model  : 

41     1  and  0  make  ^. 

-24 

2       2  and  2  make  4. 

5^     0  and  ^  make  -|. 
-2 
31     2  and  3  make  5. 

Solve: 

3a          6            c           ci 
41      5i      61      4| 
-21   -2     -31   -21 

e    '      f           g          h           i 

4|      4|      71      5|      7| 
-31  -2|  -21   -3i   -31 

4.      a               6                c 

14|       231       24| 

_6i    -18      -181    - 

d              e               f             g 
26|       301       401       45| 
.   9      -161     -10      -28J 

Find  the  sum  of  each  of  the  above  exercises. 


190  FRACTIONS  AND  DECIMALS 

279.    Written  Exercises. 

1.  From  27  subtract  13i. 

Model  :  27  Add  |  to  the  minuend.     J  and  i 

-13^  are  |.     Add  1  to  o.     4  and  o  are  7. 

13 J  1  and  1  are  2. 
Subtract : 
a        h          G  d         e  f  g  k  i 

2.  4       5       8  6       3        10       25        15        24 
2i      1|     3|  4|      2|     _4J      12|     Jl     20^ 

a  b  c  d  e 

3.  5         7         10         8         8 

a  6  c  d  e 

4.  35        37        67        34        30 
28|     _9|      18|     10|      lOi 

5.  A  woman  bought  8  yd.  of  cloth.  She  used  2^ 
yd.  in  making  a  waist.  How  much  of  the  cloth  had 
she  left  ? 

6.  A  grocer  bought  9  doz.  eggs.  He  sold  5^  dozen. 
How  many  dozen  did  he  have  left  ? 

7.  A  girl  bought  10  yards  of  lace.  She  used  3| 
yards  to  trim  a  dress.  How  many  yards  had  she 
left? 

8.  A  girl  bought  2  pieces  of  ribbon.  One  of  the 
pieces  was  6^  yards  long  and  the  other  was  4^ 
yards  long.  How  many  yards  were  there  in  the  two 
ribbons  ? 


/ 

9 

h 

9 

25 

34 

4| 

12f 

151 

/ 

9 

h 

40 

30 

40 

20| 

lor 

19| 

ADDITION  AND  SUBTRACTION  OF  FRACTIONS     191 


280.    Written  Exercises. 


1.    From  9^  subtract  4f . 


Model  •  9^         Since  the  fraction  in  the  minuend 


_42 
^3 

4 
Addl 


is  less  than  the  fraction  in  the  siib- 
2     trahend,  add  f  or  1  to  the  fraction 


^     of   the    minuend,     f   and   -| 
L  to  4.     5  and  4  are  9.^ 

are   i. 

a            h            c              d              e               f 
91-        71        81        141       22i       341 

g 
401 

-^ 

-2| 

-H 

-4f 

-15| 

-17f 

-lOf 

9| 

7| 

■    8| 

12* 

27i 

341 

17i 

-4* 

-^ 

-4f 

-3i 

-18 

-26| 

-9f 

4.  Find  the  sum  of  each  of  the  above  exercises. 

5.  From  a  piece  of  cloth  containing  7^  yd.  of  silk 
a  merchant  sold  3|-  yd.  How  many  yards  re- 
mained ? 

6.  A  grocer  bought  36  doz.  eggs.  He  sold  4^  doz. 
to  one  customer  and  3f  doz.  to  another.  How  many 
dozen  did  he  sell  to  both  ? 

7.  A  girl  bought  6^  yd.  of  lace.  She  used  2|  yd. 
to  trim  a  dress.     How  much  lace  had  she  left  ? 

*  After  the  pupils  have  become  familiar  with  this  method,  they  may 
be  taught  to  subtract  the  fraction  of  the  subtrahend  from  1,  and  to  add 
the  difference  to  the  fraction  in  the  minuend,  thus:  f  and  \  make  1.  -J- 
(the  difference)  and  \  (the  fraction  in  the  minuend)  make  f ,  the  fractional 
part  of  the  answer. 


192  FRACTIONS  AND  DECIMALS 

281.  Written  Exercises. 


a 

b 

C 

d 

e 

/ 

9 

h 

1.  51 

6| 

6i 

5f 

H 

5i 

6i 

H 

7| 

n 

5i 

2f 

n 

8^ 

5i 

5 

H 

8| 

2 

8| 

8| 

9i 

8 

9f 

H 

5f 

4 

9| 

7i 

7 

9| 

81 

2.  5f 

6| 

6* 

81 

5| 

6f 

5| 

n 

H 

3^ 

■n 

5| 

6i 

^ 

8f 

6f 

4f 

5| 

H 

6| 

8 

H 

2| 

8 

H 

8i 

H 

8| 

9 

8 

3 

9 

Subtract 

,: 

• 

3.  61 

6i 

^ 

^ 

H 

4f 

7f 

6| 

4f 

2| 

If 

If 

H 

1* 

2f 

1* 

4.  ^ 

6f 

8* 

9f 

6f 

n 

8f 

14i 

^ 

4 

H 

4i 

2* 

H 

3f 

9f 

5.  A  girl's  weight  on  June  15th  was  94|  lb.,  and 
on  August  15th  was  102^  lb.  How  much  did  she 
gain  in  two  months? 

6.  Alice  bought  12  yd.  of  lace  and  used  8f  yd.  to 
trim  a  dress.     How  much  of  the  lace  had  she  left  ? 

7.  Find  the  sum  of  2f  lb.,  6f  lb.,  and  8^  lb. 

8.  Find  the  sum  of  3^  yd.,  6|  yd.,  and  8f  yd. 


DEFINITIONS  198 

282.  Oral  Exercises. 

1.  In  the  fractions  f,  f ,  ^,  and  ^,  the  unit  of  meas- 
ure is  ^.  The  5  shows  into  how  man}^  equal  parts 
the  quantity  is  divided.  It  is  called  the  denominator 
of  the  fractions.  It  names  the  equal  parts.  The  2, 
3,  1,  and  4  tell  the  number  of  equal  parts  taken,  or 
the  number  of  times  the  unit  of  measure  is  taken. 
The  upper  term  is  called  the  numerator  of  the  fraction. 

2.  In  1^,  6  is  the  denominator.  It  shows  that  the 
quantity  is  divided  into  6  equal  parts,  or  into  sixths, 
and  that  the  unit  of  measure  is  ^.  The  numerator 
is  4.  It  tells  the  number  of  equal  parts  taken,  or  the 
number  of  tunes  the  unit  of  measure,  J,  is  taken. 

3.  A  fraction  whose  numerator  is  less  than  the 
denominator  is  called  a  proper  fraction. 

4.  A  fraction  whose  numerator  is  equal  to  or  greater 
than  the  denominator  is  called  an  improper  fraction. 

5.  Name  the  numerator,  the  denominator,  and  the 
unit  of  measure  in  the  following.  Tell  which  are 
proper  fractions :  f,  f ,  |,  f ,  |,  f ,  f ,  f,  |. 

6.  Such  numbers  as  8,  7,  4,  25,  etc.,  are  called 
integers.  When  a  number  is  composed  of  an  integer 
and  a  fraction,  it  is  called  a  mixed  number.  8|  is  a 
mixed  number.  It  is  expressed  in  two  units  of  meas- 
ure. The  8  is  expressed  in  ones,  the  |  in  fourths. 
It  may  all  be  changed  to  fourths.  There  are  f  in  1. 
In  8  there  are  ^-^-.     ^  and  J  are  ^-^. 

1st  Bk  Aritii — 13 


194  FRACTIONS   AND   DECIMALS 

REDUCTION  OF  FRACTIONS 

283.  To  change  a  mixed  number  to  an  improper 
fraction,  multiply  the  whole  numher  by  the  denomina- 
tor of  the  fraction,  add  the  numerator,  and  ivrite  the 
sum  over  the  denorninator  of  the  fraction. 

1.  Change  6|  to  an  improper  fraction. 

Model  :   5  x  6  =  30        ^^.  ^^^^^^  ^^^  ^  *^  ^^^^'' 
30  ,  4__34     multiply  it  by  5.     5  times  6 

"^^^""^      is  30. 

2.  Change  the  following  mixed  numbers  to  im- 
proper fractions:  ^,  9|,  7^,  ^,  \^,  15|,  ^. 

3.  Write  ten  mixed  numbers  and  change  them  to 
improper  fractions. 

4.  Change  to  improper  fractions :   Tf,  6|,  3f ,  4|-. 

284.  To  change  an  improper  fraction  to  a  mixed 
number,  divide  the  numerator  by  the  denominator. 

1.  Change  ^  to  a  mixed  number. 

61 
Model  :   .vo^    Divide  25  by  4. 

2.  Change  the  following  improper  fractions  to 
mixed  numbers  :  ^,  ^,  ^^  J^,  J^,  J^. 

3.  Change  to  mixed  numbers:   ^,  ^,  l|^,  ^^-. 

4.  Write  ten  improper  fractions  and  change  them 
to  mixed  numbers. 

5.  Find  the  sum  of  the  following  by  adding  the 
numerators  together.  Reduce  the  answer  to  a  mixed 
number:  |,  ^,  ^,  J^^  ^,  l 


REDUCTION   OF   FRACTIONS 


195 


285.  1.   Show  in  a  similar  way  :i  =  |,  l  =  |,i  =  ^-^, 

2.  Show  in  a  similar  way  :  i  =  f  ?  i  =  i%?  i  =  o"- 

3.  Change  to  12ths :  ^,  \,  ^,  ^. 

4.  Which  is  the  larger,  and  how  much,  \  ov  ^-^1 

5.  The  fractions  ^,  f,  f ,  and  ^^  are  alike  in  value. 
They  differ  in  /orm. 

6.  Can  you  add  the  following  fractions  as  they 
stand:  J  ft.,  |  ft.,  and  f  ft.? 

7.  Can  you  add  the  following  fractions  as  they 
stand:  1  ft.,  l  ft.,  and  l  ft.? 

8.  Can  you  add  the  fractions  in  Question  7  if  they 
are  changed  to  inches? 

9.  Can  you  add  the  following  fractions :  ;^  ft., 
^  doz.,  1  gal.?  Is  there  a  common  unit  to  which 
they  can  be  changed  ? 

10.  Multiply  the  numerator  and  the  denominator 
of  ^  by  4.  The  answer  is  — .  Has  the  value  of  the 
fraction  been  changed  ? 

11.  Multiply  the  numerator  and  the  denominator 
of  1  by  4.  The  answer  is  — .  Has  the  value  of  the 
fraction  been  changed  ? 


196  FRACTIONS   AND   DKCIMALS 

286.  Oral  Exercises. 

1.  Change  f  to  12ths. 

Model:  3  is  contained  in  12  four 

1  =  i| ;  1=1%;  f  =  3-%.     times.     4  times  2  is  8. 

2.  When  I  is  changed  to  12ths,  the  denominator  is 
made  4  times  as  large.  We  know  this  because  3  is 
contained  in  12  four  times.  The  numerator  must 
also  be  made  4  times  as  large,  so  it  is  multiplied 
by  4. 

3.  Change  to  12ths  : 

4.  Change  to  18ths  : 

5.  Change  to  24ths  : 

6.  Change  to  20ths  : 

7.  Change  to  36ths  : 
a    Change  to  30ths : 

287.  Written  Exercises. 
Change  to  12ths  and  add 

Model  A  :         Model  B  : 


h 

h 

h  h  h  h  h 

h 

h 

h  h  h  h  f. 

h 

h 

6'  h  1%  h  A- 

4 
5"' 

3 

h   iV  h  h  A- 

h 

3 
4' 

h  T2?  h  h  9- 

h 

h 

A'  A>  h  i' 

2.  3.  4. 


i=A 

* 

2 

i 

i 

l=A 

1 

8 

i 

i 

i=A 

h 

6 

i 

i 

f=A 

1 

9 

i 

i 

B  2 

t       i 


Add  the  fractions  in  Exercises  3-8,  Sec.  286. 


* 


REDUCTION   OF    FRACTIONS 


197 


288.   Written  Exercises. 

1.    Find  the  sum  of  Sf,  6|,  and  Gf. 


Change  the  fractions  to  12ths  and  add  the  sum  of 
the  fractions  to  the  sum  of  the  whole  numbers. 


Model 

A: 

Model  B: 

8f  = 

8M 

8| 

10 

6|  = 

6A 

H 

8 

6f  = 

6A 
2A 

=  221 

6| 

9 

2 

221 

H= 

=  2A  =  2i 

Change 

to  12ths  and  add 

: 

2.      3. 

4. 

5. 

6. 

7. 

a 

8f    5| 

7i 

5* 

^ 

8A 

If 

6f    61 

9f 

8f 

n 

7f 

9| 

7|    71 

2i 

7f 

8A 

6i 

7J 

H         ^ 

5f 

9A 

4f 

4f 

5f 

54    3| 

6f 

8* 

5* 

3f 

6i 

i)|    81 

7J 

7i 

H 

2| 

8f 

289.  1.  The  fraction  f  may  be  changed  to  ^  by 
dividing  the  numerator  and  denominator  by  4.  This 
does  not  change  the  value  of  the  fraction. 

2.  Remember :  If  the  numerator  and  the  denomi- 
nator of  a  fraction  are  divided  by  the  same  number, 
the  value  of  the  fraction  is  not  changed. 

3.  The  fraction  |^  is  not  in  its  lowest  terTus  because 
both  the  numerator  and  the  denominator  may  be 
divided  by  a  number  that  will  change  them  to  smaller 
numbers  without  changing  the  value  of  the  fraction. 
What  is  the  number  ? 


198  FRACTIONS  AND  DECIMALS 

MEASURES 

290.  1.    The  exact  measures  of  12  ft.  are :    1  ft., 

2  ft.,  3  ft.,  4  ft.,  6  ft.,  and  12  ft. 

2.  The  exact  measures  of  18  ft.  are :  1  ft.,  2  ft., 

3  ft.,  6  ft.,  9  ft.,  and  18  ft. 

3.  2  ft.,  3  ft.,  and  6  ft.  are  each  exact  measures 
of  12  ft.  and  18  ft.  They  are  common  measures  of 
12  ft.  and  18  ft. 

4.  6  ft.  is  the  greatest  measure  that  is  common  to 
12  ft.  and  18  ft.  It  is  the  greatest  common  measure 
of  12  ft.  and  18  ft. 

291.  Find  the  exact  measures  of: 

1.  15  gal.       4.    16  qt.  7.    $26.       lo.    18  in. 

2.  20  gal.       5.    24  pt.         a    $30.      ii.    28  da. 

3.  36  ft.         6.    40  gal.        9.    $48.       12.    10  ft. 

Find  the  common  measures  of : 

13.  12  and  18.       16.    16  and  36.  19.  24  and  36. 

14.  24  and  30.       17.    10  and  40.  20  30  and  36. 

15.  36  and  48.       la    12  and  48.  21.  14  and  28. 

292.  Keduce  to  lowest  terms  ; 

1        6_  \2.0     JL    14     3  2.    16.  A      2J.     42'3_^     30     40     6  4 

•*••     36?   36?   36?   36'   36?   36*         *'     36'   6lF?   42?   4ir?   64?    T'J* 

32     14     1  « 

3^6?  49?  "?rr- 

48      16     24 
¥9?  T2?  ¥4?  4^- 

7-  What  is  meant  by  the  greatest  common  measure 
of  two  or  more  quantities  ? 

Til  is  is  generally  known  as  the  greatest  common 
divisor,  or  greatest  common  factor  of  the  quantities. 


o  6  4_'   18.     2  0.    12.1_6  q  8       6  3  54     32  14     18 

-=•  24'  24?   24?   24'   24'   24*  ^'  24'  "ST?  63?  3^6?  4  9?  IT 

«  12  3  2     40   ^'3  6     16     44  ^  8       2  1  2  8     4  8  16      2 

3-  4ir?  4¥?  ¥¥?  ¥¥'  4¥'  4¥-  ^  48'   SS"? 


MEASURES 

199 

293.   Oral  Exercises. 

— 

Change 

!  to  improper  fractions : 

1. 

3f 

6.     9f             11. 

^    16. 

6f 

21.    8J, 

2. 

5| 

7.     8|             12. 

9|             17. 

5| 

22.     7^2- 

3. 

6i 

8.     5J             13. 

8i           18. 

2| 

23.     63^ 

4. 

7f 

9.     6f             14. 

4f            19. 

5| 

24.     SjJj 

5. 

8f 

10.     7|             15. 

3f            20. 

7i 

25.     8A 

294.   Oral  Exercises. 

Change 

!  to  whole  or  mixed  numbers : 

1. 

¥  ft. 

6.  \«  lb.    11. 

J^ft.       16. 

fft. 

21.   %^^ 

2. 

yin. 

7.  ^qt    12. 

.2;tda.     17. 

|yd. 

22,    $V- 

3. 

^da. 

8.  Jf  ft.     13. 

^  mi.    18. 

V  in. 

23.    $J# 

4. 

¥yd. 

9.  -2^  yd.   14. 

¥pt.    19- 

J^mi 

.      24.    1-3^ 

5. 

-V-  da. 

10.  y  rd.    15. 

J^  lb.      20. 

¥ft- 

25.    $¥ 

295.   Oral  Exercises. 

1. 

ift.  = 

A  ft.       5.    1   ' 

ia.  =  f^  da. 

9.   f 

ft.  =  f^ft. 

2. 

|y«i.= 

i'2  yd.  6.  1  ( 

ia.  =  2°4  da. 

10.    t 

ft.  =  ^ft. 

3. 

f  mi.= 

f^mi.    '•    f  ' 

3a.  =  ^  da. 

"•A 

ft.  =  A  ft. 

4. 

ift.  = 

^ft.      8.    1  ( 

ia.  =  ^  da. 

12.    1 

ft.  = /oft. 

296.    Oral  Exercises. 
Reduce  to  lowest  terms : 


1.  ^ 
2. 


3. 


M 

5. 

24 
60 

9. 

1  0 
120 

13. 

90 
900 

17. 

M 

21. 

\\ 

li 

6. 

fl 

10. 

M 

14. 

e 

18. 

W 

22. 

tVi 

f^ 

* 
7. 

T2¥ 

11. 

f* 

15. 

M 

19. 

M 

23. 

10  8 

M 

8. 

ro7 

12. 

ft 

16. 

M 

20. 

4  4 
55 

24. 

^ 

200  FRACTIONS  AND  DECIMALS 

FACTORS   AND    MULTIPLES 

297.    Oral  Exercises. 

1.  Numbers  that  are  exactly  divisible  by  2  are 
even  numbers. 

2.  Name  the  even  numbers  to  30.  Numbers  that 
are  exactly  divisible  by  2  end  in  what  figures  ? 

3.  Numbers  that  *  are  not  exactly  divisible  by  2 
are  odd  numbers. 

4.  Name  the  odd  numbers  to  30. 

5.  Some  numbers  cannot  be  divided  by  any  whole 
numbers  except  themselves  and  1  without  leaving 
a  remainder.  These  are  called  prime  numbers.  The 
following  are  prime  numbers :  1,  2,  3,  5,  7,  11, 13,  17, 
19,  23,  29,  31,  37,  41,  43,  47,  53,  59,  61,  67,  71,  73, 
79,  83,  89,  97.  These  can  be  divided  only  by  them- 
selves and  1  without  leaving  a  remainder. 

6.  A  factor  of  a  number  is  one  of  two  or  more 
numbers  which  multiplied  together  will  make  the 
number.  6  and  5  are  each  factors  of  30.  2,  3,  4,  and 
6  are  each  factors  of  12. 

7.  A  common  factor  is  a  common  unit  of  measure. 
6  is  a  common  factor  of  12  and  of  36,  because  both 
of  these  numbers  are  exactly  divisible  by  6.  Name 
the  common  factors  of  24  and  36. 

To  reduce  a  fraction  to  its  lowest  terms,  divide 
hot] I  numerator  and  denoyninator  by  their  common 
factors. 


FACTORS   AND  MULTIPLES  201 

298.  Oral  Exercises. 

1.  The  multiples  of  2  are :  2,  4,  6,  8,  10,  12,  etc. 

2.  The  multiples  of  3  are  :  3,  6,  9,  12,  15,  etc. 

3.  Which  of  the  above  numbers  are  multiples  of 
both  2  and  3  ? 

4.  Numbers  that  are  multiples  of  two  or  more 
numbers  are  called  common  multiples  of  the  numbers. 
6  and  12  are  common  multiples  of  2  and  3. 

5.  The  least  common  multiple  of  2  and  3  is  6. 

6.  Find  the  least  common  multiple  of  3  and  5. 

7.  Find  the  least  common  multiple  of  6  and  8. 

8.  Find  the  least  common  multiple  of  6  and  9. 

9.  In  the  fractions  f  and  f ,  the  least  common  mul- 
tiple of  the  denominators  is  12.  It  is  called  the  least 
common  denominator  of  the  fractions. 

299.  Written  Exercises. 

1.  Change  to  least  common  denominators  and  add : 
abcdefgh.ijk  I 


2 

4 

5 

4 

1 

3 

2 

2 

7 

1 

4 

5 

3 

5 

6 

7 

2 

8 

9 

5 

8 

9 

5 

T2 

3 

3. 

2. 

3. 

3. 

5 

1 

5 

3. 

3. 

JL 

3. 

£ 

^ 

3^ 

± 

A 

_6^ 

^ 

6_ 

± 

± 

_8^ 

4 

2.  What'  i^tne' least  common  multiple  of   2,  3, 
and  4? 

3.  What  is  the  least  common  multiple  of   3,  4, 
and  5? 

4.  What  is  the  least  common  multiple  of    6,  4, 
and  8? 


202  FRACTIONS  AND  DECIMALS 

5.  Name  the  prime  numbers  below  10. 

6.  Find  the  least  common  multiple  of  7  and  9. 
Since  7  is  a  prime  number,  the  least  common  multiple 
of  7  and  9  is  their  product. 

7.  What  is  the  least  common  multiple  of  7  and 
8  ?     Of  7  and  6  ?     Of  5  and  7  ?    Of  5  and  9  ? 

300.   Written  Exercises. 

1.    From  5f  subtract  2|-. 

Model  A :    5f  =  5^^  Model  B  :    5f  1 8 

-2|  =  2^  -2||9 

Add  ^  to  the  fraction  of  the  minuend,  making  it 
f  f .     ^  and  ^  are  ff .     Add  1  to  2.     3  and  2  are  5. 

Solve : 
abcdefgh 

2.    6f       5|       6i       7|       8i       6f       81       9| 
-41    -2|    -41    -5i    -5|    -4|    -4§    -6^ 


3.15^      19f      ISf        9f      121       7^^  6|-       5h 

_8i    -9|    -7f    -4^    -8^-21  -4A-4f 

4.  18f      16f      28        43f      26f     30  30|     SOf 

-S^.    -4i    -7|     -7_    -4f    -^1  -9^    -9f 

Add  each  of  the  above  exercises. 


ORAL  EXERCISES  203 

301.  Oral  Exercises. 

1.  Change  to  improper  fractions  :   3f ,  8|,  9f ,  7f . 

2.  Change  to  whole  or  mixed  numbers :   -^^  -^, 

23      60. 

7  ?  '9  • 

3.  Reduce  to  lowest  terms  :  ff ,  If?  it?  if?  f  p 

4.  What  are  the  common  factors  of  12  and  8  ? 

5.  What  is  the  least  common  multiple  of  12  and  8  ? 

6.  Change  |  to  30ths  ;  |  to  40ths  ;  f  to  24ths. 

n       !_»'_.     !—_£_.     3. .__»!_.    1  —  _£L_  •      2  __      a-      . 

'•      2~T0  0?    4"~10  0'    4~1005    5""100?     5"~100> 

3  3!      .4  g 

;S""T0(5  '    5  ""  TWO' 

8.  Findfof  $12;  fof  $12;  fof  $12;  |of24hr. 

9.  6  is  fof—;  8  isfof  — ;  9  is  fof—;    12  is 
fof—. 

10.    Name  the  prime  numbers  between  1  and  30. 

302.  Written  Problems. 

1.  A  girl   bought   8|  yd.  of  ribbon.     She  used 
5f  yd.     How  many  yards  had  she  left  ? 

2.  James  weighs  84f  lb.  and  George  weighs  97  lb. 
How  much  heavier  is  George  than  James  ? 

3.  What  is  the  sum  of  6|  yd.  and  8f  yd.  ? 

4.  A  man  owned  247f  acres  of  land.     He  sold 
122|^  acres.     How  many  acres  had  he  left  ? 

-  5.    The  sides  of  a  field  are  271|  rd.,  290  rd.,  175  f 
rd.,  and  180 J  rd.     Find  the  distance  around  the  field. 
6.    Find  the  distance  around  a  room  that  is  14J  ft. 
long  and  9|  ft.  wide. 


204  FRACTIONS   AND  DECIMALS 

303.   Written  Exercises. 

Add: 


a 

6 

c 

d 

137^ 

66f 

997J 

54^ 

142tV 

88| 

885| 

68,^0 

183/^ 

99j^ 

6673-V 

88,V 

988| 

88* 

832| 

99A 

ST'JtV 

991 

238^3^ 

65M 

777H 

78A 

9651 

76f 

304.    1.    Harry  weighs  72^  lb.     George  weighs  2| 
lb.  more  than  Harry.     Find  the  weight  of  George. 

2.  Find  by  addition  the  cost  of  8  yards  of  cloth 
at  12|^  a  yard. 

3.  A  tailor  had  a  piece  of  cloth  containing  24|  yd. 
From  this  piece  he  used  3f  yd.  to  make  a  pair  of 
trousers.     How  much  of  the  piece  remained  ? 

4.  A  grocer  bought  sugar -at  4f  ^  a  pound.  He 
sold  it  at  the  rate  of  18  lb.  for  $1.  How  much  did 
lie  receive  per  pound  for  it  ?   Find  his  profit  per  pound. 

5.  At  14^^  a  pound,  what  is  the  cost  of  a  turkey 
that  weighs  8  pounds  ? 

6.  At  12^^  a  pound,  what  is  the  cost  of  a  roast 
that  weighs  8  pounds  ? 

7.  After  selling  3^  yd.  of  ribbon,  there  was  left 
9|  yd.     Find  the  length  of  the  piece  before  the  sale. 

8.  What  number  subtracted  from  12  leaves  9  ? 
What  number  subtracted  from  |-  leaves  J  ? 


'J^ 


WRITTEN   EXERCISES 


205 


305.   Written  Exercises. 

1.  Reduce  to  a  common  denominator  :  ^{  f ,  |^,  j^q  . 
10  is  a  multiple  of  5 ;  4  is  a  multiple  of  2.  There- 
fore, we  need  to  find  only  the  common  denominator 
of  4  and  10.  This  will  contain  all  of  the  denomi- 
nators.    Why  ? 

2.  Reduce  to  a  common  denominator  and  add. 
First  determine  which  denominators  need  not  be 
considered. 


/ 


1  i 

f  \\ 

5 
9 

\ 

«j 

^1 

A   i 

\ 

f  f 

4    i 

i  * 

i 

f 

t ! 

1 

1   1 

i 

if  f 

1    1 

3    3  * 
8   10 

3    4 

1  > 

4 

1 

f   f 

i 

2   10 

5    7 

1    5 

1    2 

1 

3 

1    5 

5 

3    1 

6    9 

2    6 

2    3 

3 

4 

3     6 

12 

7    2 

\%        1%. 

306. 

M:  V 

^ 

1 

a 

6 

c 

a 

e 

/ 

9- 

24f 

67i 

34| 

761 

691 

58f 

791 

661 

53|- 

65* 

84f 

73f 

741 

83* 

731 

49| 

47A 

59f 

8Vo 

561 

95| 

94f 

78tV 
Subtra( 

68f 

72| 

961 

871 

82tV 

307. 

3t: 

a. " 

Jl; 

£." 

^, 

d^   X 

JL- 

/» 

1.  78| 

96J 

67f 

86| 

90f  65| 

98f 

43A 

-25| 

'80f  -37f  - 

.47|. 

-37f  -24| 

-54| 

-17| 

2.  96f 

744 

28f 

79A 

851  47| 

781 

98t'2 

47* 

36f 

lOf 

64f 

26f  16f 

45f 

12| 

206  FRACTIONS  AND  DECIMALS 

MULTIPLICATION   OF   FRACTIONS 

308.   Oral  Exercises. 

1.  In  tlie  fraction  |,  which  figure  tells  the  size  of 
the  parts?     What  does  the  numerator  tell ? 

2.  How  does  the  fractional  part  f  compare  in 
size  with  the  fractional  part  ^  ? 

3.^  What  is  the  sum  of  f  and  f  ?     Of  |  and  |  ? 

4.    What  is  the 'sum  of  f  and  f  and  f1    f  =  —  ^. 

9.    What  is  3  times  f  ?     What  is  3  times  |  ? 

To  multiply  a  fraction  by  a  whole  number^  multiply 
its  numerator  by  the  whole  number.  If  the  product 
is  an  iynproper  fraction,  reduce  it  to  a  ivhole  or  a 
mixed  number. 

6.  Multiply :  I  by  5,  f  by  6,  f  by  4,  f  by  3. 

7.  Which  represents  the  larger  fractional  part, 
iori?     iorl?     lori?     l  or  i?     lor^? 

8.  How  does  the  length  of  ^  ft.  compare  with  the 
length  of  i  ft.  ?    i  yd.  with  |  yd.  ?    i  ft.  with  j\  ft.  ? 

9.  If  the  denominator  of  J  is  divided  by  2,  the 
fraction  is  changed  to  ^.  |^  is  2  times  J.  To  mul- 
tiply J  by  2,  divide  its  denominator  by  2. 

To  multiply  a  fraction  by  a  whole  number,  divide  its 
denominator  by  the  whole  number  if  the  denominator 
is  exactly  divisible  by  the  whole  mimber.  If  the  residt 
is  an  improper  fraction,  change  it  to  a  whole  or  a 
mixed  number. 

10.  Multiply  by  dividing  the  denominator :  J  by  8 ; 
tby3;  l|by9;eby6;Mby7. 


MULTIPLICATION   OF   FRACTIONS  207 

11.  Multiply  ^1  by  9  by  multiplying  the  numerator 
by  9. 

12.  Multiply  ^1^  by  9  by  dividing  the  denominator 
by  9. 

13.  Which  of  the  above  methods  is  the  easier? 
Why? 

14.  When  can  the  easier  method  be  used  ? 

309.  Oral  Exercises. 

Wherever  possible,  divide  the  denominator. 

abed  e  f 

1.  fx5     fx3     |x3     3^x6     {^x7     ^x4 

2.  fx3     |x3     |x7     ^x5     ^x6     ||x8 

3.  |x4     |x4     |x8     ifx7     ^x9     ||x4 

310.  Oral  Exercises. 

1.  What  is  the  meaning  of  4x2?    4x1?    4x1? 

2.  4x|  is  the  same  as  ^  of  4.    ^   of  4  =  — . 
4x1=-. 

3.  12  X  i  is  the  same  as  |^  of  12.     12  x  |  =  —  . 

4.  If  12  is  multiplied  by  f,  will  the  answer  be 
greater  or  less  than  12  ? 


5.    Multiply  12  by 


Model  :  l  of  12  is  2 ;  f  of  12  are  5  times  2,  or  10. 

6.    Multiply:  18byf;  14byf;  12byf;  16by|; 

18  by  I;   10  by  f ;   24  by  |;   16  by  |;    18  by  |; 

16  by  ff;   20  by  |;   24  by  f ;   30  by  ^\;  25  by  f ; 

2 
3 


12  by  ^ 


208  FRACTIONS   AND  DECIMALS 

311.    Oral  Exercises. 

1.   ^  of  7  may  be  indicated  thus:  -|.     This  is  read 
seven  thirds,  or  7  divided  by  3. 


2.   1  of  10  =  V*-,  or  3f     J  of  9  =  f ,  or 


1 

4- 


3.  i  of  5  =  J.     iof9  =  f.     ^of4  =  f.     iof5  =  f. 

4.  Find  f  of  7. 

Model  :  i  of  7  =  21      f  of  7  =  2  times  21,  or  4f . 

5.  Solve:  9x|;  8xf;  11  xf;  8xf;  4xf. 

To  multiply  a  whole  number  by  a  fraction,  divide 
the  'whole  number  by  the  denoininator  of  the  fraction 
and  mtdtiply  the  quotient  by  the  numerator. 

312.    Oral  Exercises. 


a 

6 

c 

d 

e 

1.    9x|- 

5x| 

llxf 

18  xi 

12  x| 

2.    7x| 

6xf 

24  xf 

11  x| 

27x| 

3.   8xf 

16  xf 

12  xf 

12  x^ 

21xf 

4.   7xf 

18  x| 

18  xf 

30  xf 

22  x,,*^ 

5.   6x| 

20xf 

12  xf 

16x| 

40  X  ^% 

313.   Oral  Exercises. 

a 

6 

c 

d 

e 

1.   |of30 

|of7 

T.%of60 

Hx9 

20xf 

2.   |of40 

|of5 

Wof50 

Ax6 

15xf 

3.  |of35 

fofS 

|x7 

Mx6 

25xt 

4.    7  of  56 

fof3 

4x^ 

21  xf 

12xf 

5.  |of24 

f  of  49 

fx7 

18x1 

11  xf 

MULTIPLICATION  OF   FRACTIONS 


209 


314.    Written  Exercises. 


1.   Multiply  24|  by  8. 
Model  :     24| 

8 

6 

192 

198 

First,  multiply  |  by  8.     8 
times  1  =  -^^-  =  6 .     Next,  multi- 
ply 24  by  8.     Add  the  products. 

a 
2.   35fx5 

b 
345|  X  4 

c 

3065x75 

d 
725| X  84 

3.   6T|x8 

725|  X  3 

937f  X  84 

423|x60 

4.   95fx5 

467f  X  4 

7841 X  23 

596| X  70 

5.    301x6 

879f  x8 

986| X  46 

640| X  50 

315.    Written  Exercises. 

1.    Multiply  64  by  4f.. 
Model  :     64 


First,  multiply  64  by 


4| 
38f 
.256' 
294| 
Solve : 

f.     lof64=12i      fof 
64  =  3  times  12^  or  38f . 
Next,  multiply  64  by  4. 
Add  the  products. 

a 
2.   675  x9f 

6 

300  X 

4f 

c 
464  X  6f 

d 
723  X  6| 

3.   864  x4f 

950  X 

2f 

405  X  91 

800  X  5| 

4.   576  X  5| 

375  X 

8f 

672  x3| 

967  X  6| 

5.    674  x8|- 

675  X 

3f 

456  X  51 

734  X  4f 

IST  I'.K  AKMII  — 14 

210 


FRACTIONS   AND   DECIMALS 


316.   Written  Exercises. 

Multiply 

each  by  6, 

8,  and  9: 

a 

h 

c 

d 

e 

1.    23  If 

579| 

768-1 

756| 

697f 

2.    327f 

968f 

648f 

748^ 

764f 

3.   432f 

786f 

975^ 

654H 

924f 

317.    Oral  Exercises; 
Red  ace  to  lowest  terms : 


if 

18 
24 

20 
3-0 


14 

28 

27 
36" 

in 

48 


3.6 

48 

M 

64 

T2 


d 

12 
144 

19 
TF8 

56 
96 


14  4 

50 
T70 

20 


f 

25 
TFCT 

40 
IFT) 

1^ 


318.  Oral  Exercises. 
Change  to  wliole  or  mixed  numbers : 

1. 

2. 

3. 

319.  Written  Exercises. 

Find  the  sum  and  the  difference  of  each : 


a 

b 

C 

d 

e 

/ 

!/ 

¥- 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

100 

¥ 

¥ 

¥ 

¥ 

fl 

ft 

100 

¥ 

¥ 

W 

1. 


97|  lb. 
28^  lb. 

104f  ft. 
74i  ft. 

924f  lb. 
634^  lb. 

764j3g  mi. 
684,-V  mi. 

76^  mi. 
241  mi. 

367f  A. 
145f  A. 

375fyd 
194^  yd. 

6941  lb. 
375f  lb. 

WRITTEN   PROBLEMS  211 

320.    Written  Problems. 

1.  Find  the  cost  of  6  lb.  of  sugar  at  5|^  a  pound. 

2.  At  12^^  a  pound,  how  much  will  6  pounds  of 
meat  cost? 

3.  What  is  the  cost  of  3|-  lb.  of  steak  at  16^  a 
pound  ? 

4.  George  lives  4f  mi.  from  the  city.  How  far 
must  he  ride  in  making  2  round  trips  to  the  city  ? 

5.  The  average  weight  of  4  boys  is  87f  lb.  Find 
their  total  weight. 

6.  Alice  bought  51-  yd.  of  lace.  She  used  2|  yd. 
to  trim  a  waist.     How  much  of  the  lace  had  she  left  ? 

7.  If  each  can  of  milk  contains  of  gal.,  how  much 
milk  is  there  in  6  cans  ? 

8.  A  man  walked  at  the  rate  of  4  miles  an  hour. 
It  took  him  2|  hr.  to  go  from  his  home  to  the  city. 
How  far  from  the  city  did  he  live  ? 

9.  A  man  had  137i  A.  of  land.  He  sold  43f  A. 
to  one  man  and  641  A.  to  another.  How  much  did 
he  sell  to  both  ?     How  many  acres  had  he  left  ? 

10.  How  much  will  2|  lb.  of  tea  cost  at  60^  per 
pound? 

11.  Find  the  cost  of  -  8  yd.  of  cloth  at  121^  per 
yard. 

12.  How  many  rods  of  fence  will  it  take  to  fence  in 
a  garden  14 1  rd.  long  and  8|  rd.  wide? 

13.  Find  the  cost  of  8f  T.  of  coal  at  |  6  per  ton. 


212  FRACTIONS  AND  DECIMALS 

DIVISION  OF  FRACTIONS 

321.    Oral  Exercises. 

1.  How  does  the  fractional  part  |  compare  in  size 
with  the  fractional  part  f  ? 

2.  How  can  the  fractional  part  f  be  obtained  from 
the  fractional  part  ^  ? 

3.  How  can  the  fractional  part  |  be  obtained  from 
the  fractional  part  f  ? 

4.  If  the  fractional  part  f  is  divided  by  2,  the 
quotient  will  be  y. 

To  divide  a  fraction  hy  a  ivhole  number,  divide  its 
numerator  hy  the  whole  number. 

5.  Divide  f  by  2;  ^  by  2;^f  by  2;  ||  by  3; 
f  by  3. 

6.  Divide  ff  by  7;  f  by  4;  f  by  2;  ff  by  5;  |f 
by  6  ;  ^1  by  4. 

7.  Divide  4  by  4;  If  by  10;  If  by  8 ;  ,-%  by  3; 
H  by  3;  1^  by  5. 

8.  How  does  the  fractional  part  J  compare  with 
the  fractional  part  J? 

9.  What  part  of  J  is  |  ?     What  part  of  J  is  f-^  ? 

10.  How  does  the  fractional  part  ^  compare  with 
the  fractional  part  i  ?  How  can  the  fractional  part  ^ 
be  obtained  from  the  fractional  part  ^  ? 

11.  If  the  fractional  part  ^  is  divided  by  2,  the 
quotient  will  be  — .  Multiplying  the  denominator  of 
a  fraction  by  2  has  what  effect  upon  the  value  of  the 
fraction  ? 


DIVISION  OF  FRACTIONS  213 

322.  To  divide  a  fraction  hy  a  ivliole  number^  mid- 
tq^ly  the  denominatoi'  of  the  fraction  hy  the  ivhole 
number. 

1.  Divide:  I  by  4;  I  by  5;  f  by  7;  |  by  2;  f  by  5. 

2.  Divide:  i  by  2;  f  by  3;  I  by  3;  |  by  4;  i  by 
6;  iby  5. 

3.  Divide:  I  by  5;  11  by  2;  f  by  4;  |  by  3; 
k  by  3. 

4.  Divide  the  fraction  i|  by  6  by  dividing  its 
numerator  by  6.  To  divide  by  6  is  to  find  ^.  Find 
1  of  ^^ 

5.  Divide  the  fraction  ^  by  6  by  multiplying  its 
denominator  by  6.     Reduce  to  lowest  terms. 

6.  Which  is  the  easier  method  of  dividing  i|  by 
6?     Why? 

7.  When  is  it  easier  to  find  the  quotient  by  dividing 
the  numerator  by  the  whole  number  ? 

8.  When  is  it  easier  to  find  the  quotient  by  multi- 
plying the  denominator  by  the  whole  number? 

323.  Oral  Exercises. 

Use  the  easier  method  in  solving  each : 


1. 


a 

6 

C 

d 

e 

f-^3 

1-4 

H-5 

H-6 

tt-12 

1-3 

1^10 

lf-6 

H-7 

M-25 

1-5 

f-10 

lf-2 

A-3 

tVo-10 

1^3 

H-7 

If -5 

A-4 

tVo-20 

214  FRACTIONS   AND   DECIMALS 


324.    Oral  Exercises. 

Use  the 

easier  method  in  sol 

ving  each : 

a 

h 

C 

d 

e 

1^5 

f-^4 

il-5 

30  X  fo 

iof72 

fx3 

9xf 

il-8 

fof7 

f  of27 

|x4 

.8xf 

lf-5 

|of2 

1  of  25 

|x4 

Ax3 

H-5 

f  of  9 

foflO 

i-6 

1-^x5 

17x1 

|of5 

A  of  8 

f-6 

M-5 

lOxf 

|of48 

A  of  50 

1-6 

M-8 

27  x| 

|of  12 

Mof  24 

325.   Written  Exercises. 

1.  Find 

i  of  212f . 

Mnn-pT. ' 

.     Jm. 

fi  is  pnnf 

ainpri  in  /^,1  f 

lirpp  f  iiTiPs 

witli  3  remainder.  6  is  contained  in  32  five  times, 
with  2  remainder.  The  whole  remainder  is  2|.  Re- 
duce it  to  ^.  ^  -5-  6  =  JJ.  This  is  the  fractional 
part  of  the  quotient. 

2.  Divide  each  by  6  ;  by  7  ;  by  4  : 

632f,  3451   426f,  7851,  967f,  872f 

326.    Oral  Exercises. 

Divide  each  l)y  4 ;  by  5 ;  by  6  ;  by  7  ;  by  8;  by  9 : 


a 

6 

f? 

rf 

e 

1.    4| 

6! 

9f 

7§ 

lOf 

2.     61 

8i 

8f 

9f 

12i 

3.    8| 

^ 

5i 

8f 

lit 

4.    7| 

2f 

7§ 

n 

1% 

WRITTEN   PROBLEMS  215 

a27.   Written  Problems. 

1.  Find  the  cost  of  2|  lb.  of  coffee  at  30^  per 
pound. 

2.  A  boy  had  45^.  He  spent  f  of  his  money  for 
a  book.     Find  the  cost  of  the  book. 

3.  A  horse  was  bought  for  $120  and  sold  for  1| 
times  its  cost.     Find  the  selling  price  of  the  horse. 

4.  There  are  60  minutes  in  one  hour.  How  many 
minutes  are  there  in  ^  of  an  hour  ? 

5.  Find  the  area  of  a  blackboard  lOf  ft.  long  and 
3  ft.  wide. 

6.  A  blackboard  containing  38J  sq.  ft.  is  3  ft.  wide. 
Find  its  length. 

7.  Six  girls  bought  a  box  of  candy  weighing  1^  lb. 
They  shared  it  equally.  How  much  candy  did  each 
girl  receive  ? 

8.  Mary  is  in  school  5^  hr.  each  day.  How  many 
hours  is  she  in  school  each  week  ? 

9.  A  woman  bought  12  yd.  of  silk  at  %  IJ  per 
yard.  She  handed  the  dealer  $25.  How  much 
change  should  she  receive  ? 

10.  A  girl  had  6^  yd.  of  ribbon.  After  using  f  yd. 
for  a  bow,  how  much  had  she  left  ? 

11.  At  12i^  a  dozen,  how  much  will  8  dozen  eggs 
031  ? 

12.  Find  the  area  of  a  field  53^^  rd.  lono^  and  30  rd. 
wide.  There  are  160  sq.  rd.  in  1  A.  Find  the 
number  of  acres  there  are  in  the  field. 


216  FRACTIONS  AND  DECIMALS 

328.  1.  Draw  an  oblong.  Divide  it  into  4  equal 
parts.     What  is  each  part  called  ? 

2.  Show  f  of  the  oblong.  Show  J  of  |^  of  the  oblong. 
■^  of  f  of  the  oblong  is  what  part  of  the  oblong  ? 

3.  Show  f  of  f  of  the  oblong.  |  of  f  of  the  oblong 
is  what  part  of  the  oblong  ? 

4.  J  of  f  is  the  same  as  f  x  ^,  which  is  read  J 
multiplied  by  J.     It  is  equal  to  — . 

5.  f  of  f  is  the  same  as  |^  x  |.     It  is  equal  to  — . 

To  multiply  a  fraction  hy  a  fraction,  multijoly  the 
numerators  together.  This  product  is  the  numerator 
of  the  answer.  Multiply  the  denominators  together. 
This  product  is  the  denominator  of  the  answer.  The 
answer  shoidd  be  expressed  in  its  lowest  terms, 

6.  Multiply  f  by  |. 


Model:  fxi    r.^% 


2x4=^ 
3x5=15* 

7.  Multiply  f  by  f ;  f  by  I;  f  by  f ;  f  by  f. 

8.  Multiply  f  by  f.  If  we  multiply  as  above,  the 
answer  is  -^.  To  reduce  -f^  to  its  lowest  terms,  both 
the  numerator  and  the  denominator  must  be  divided 
by  3.  The  answer  is  f.  Since  there  is  a  3  in  one  of 
the  numerators,  and  a  3  in  one  of  the  denominators, 
the  reduction  can  take  place  before  the  multiplication. 

1 

no        f) 

thus :  -  X  ^  =  ^-     This  is  called  cancellation. 
P     5     5 

1 


MULTIPLICATION    OF   FRACTIONS 


217 


329.    Oral  Exercises. 
1.    Multiply  f  by  f 

^  Cancel  the  common  factors. 

Divide  the  5  in  ^  by  5,  and  the 
5  in  f  by  5.     Multiply. 


Model  : 

4      r^      4 

1 

a 

b 

2. 

|xf 

i*x| 

3. 

ixf 

lfx| 

4. 

8^2 
9^3 

i-fxf 

X 


6 


2ii 
21 


11 
13 


5^12 
■8^13 

T  ^  12 

9     V   7 
T3"^  9 

9     y   5 


12  ^9 
14  y  5 

4    V  2 
11^  3 


13 


TO -^^S" 


X 


9  '^  8 

5  V  2 
6"'^  3 


3  V  1 
4^8 

1  x-S. 


330.    Written  Exercises. 
1.    Multiply  6f  by  4- 


3 
4- 


Model 


Solve : 


62 
^3 

4* 


2^ 

^3 

24 
31| 


iof  6f  = 
If,  or  5. 
4  times  6 
nets. 


If- 


f  of  6f  is  3  times 


4  times  #  is  #,  or  2|. 


3? 


is  24.     Add  these  prod- 


5. 


3|x6i 

9|x8f 

4fx4| 

4|x6f 

5|x6f 

5fx8f 

6Jx4i 

9fx7| 

9f  x  3| 
4fx3f 
5i  X  71 


lit  X  3,-^ 

m  X  4f 
19fx4f 

lOf  X  4| 


218  FRACTIONS   AND   DECIMALS 

MEASUREMENTS 

331.  1.  Draw  a  line  4  ft.  long.  Measure  it  with  a 
measure  2  ft.  long.  How  many  times  did  you  apply 
the  measure  ?     4  ft.  -h  2  ft.  =  — . 

2.  Measure  a  line  4  ft.  long  with  a  measure  1  ft. 
long.  How  many  times  did  you  apply  the  measure  ? 
4ft.-^l  ft.  =  — . 

3.  Measure  a  line  4  ft.  long  with  a  measure  i  ft. 
long.  How  many  times  did  you  apply  the  measiu-e  ? 
4ft.^ift.  =  — . 

4.  Measure  a  line  8  ft.  long  with  a  measure  2  ft. 
long.  How  many  times  did  you  apply  the  measure  ? 
8  ft.^2ft.  =  — . 

5.  Measure  a  line  8  ft.  long  with  a  measure  -J-  ft. 
long.  How  many  times  did  you  apply  the  measure? 
Sft.-f-i  ft.  =  — . 

6.  Divide  8  ft.  by  2  ft. ;  by  1  ft. ;  by  1  ft. 

7. .  Measure  6  ft.  with  a  3-ft.  measure ;  with  a  1-ft. 
measure ;  with  a  |-ft.  measure ;  with  a  f-ft.  measure. 
How  many  times  did  you  apply  each  ? 

8  How  does  the  number  of  times  that  you  applied 
the  1-ft.  measure  compare  with  the  number  of  times 
that  you  applied  the  ^-ft.  measure  ? 

9.  If  the  1-ft.  measure  is  applied  6  times  in  measur- 
ing the  length  of  a  line,  how  many  times  must  the  J-ft. 
measure  be  applied  to  measure  the  same  distance  ? 

10.  How  many  times  must  the  measure  ^  ft.  be 
applied  to  measure  6  ft.  ?     6  ft. -^^  ft.  =  24. 


ORAL   EXERCISES  219 

332.  Oral  Exercises. 

1.  In  8  ft.  there  are  16  half  feet.     8  ft.  ^  1  ft.  =  16. 

2.  In  6  ft.  there  are  18  third  feet.     6  f t.  -^-  ^  ft.  =  18. 

3.  In  6  ft.  there  are  18  third  feet  or  9  two-thirds 
feet. 

4.  How  many  fourths  are  there  in  1  ?     In  8  ? 

5.  How  many  thirds  are  there  in  1  ?    In  6  ?    In  9  ? 

6.  12ft.H-l  ft.  =  — ;  12ft.-^ift.  =  — 5  12ft.^| 
ft.  =  — . 

333.  To   divide   a   lohole  numher   hy   a  fraction, 
invert  the  divisor  and  multij)hj. 


1.   Divide  12  by  | 

■. 

^      4 

d 

Model  : 

Invert  f 

tof. 

a 

b 

c 

e 

2. 

12^1 

7-f 

36  ^f 

12mi.^f  mi. 

20-f 

3. 

lo^f 

8-t 

10^1 

8  ft.  ^f  ft. 

15-f 

4. 

18^1 

6-f 

30-1 

101b.  -^flb. 

8^1 

5. 

16^1 

9H-f 

8-f 

16hr.^f  hr. 

7-t 

334.  1.  How  many  boxes  of  candy  each  weighing 
1^  lb.  can  be  filled  from  a  pail  containing  16  pounds  ? 

2.  A  girl  bought  3  yd.  of  ribbon  at  12^  a  yard. 
How  many  hair  ribbons  each  f  yd.  long  can  be  made 
from  it  ?     Find  the  cost  of  each  ribbon. 

3.  How  many  rolls  of  butter  each  weighing  i  lb. 
can  be  made  from  14  pounds  ? 

4.  There  are  5|  yd.  in  a  rod.  How  many  yards 
are  there  in  6  rods  ? 


220  FRACTIONS  AND  DPXIMALS 

335.  1.  Measure  1  ft.  with  ^-ft.  measure.  How 
many  times  did  you  apply  the  measure  ?     1  f  t.  h-  1  ft.  =  4. 

2.  Measure  |^  ft.  with  ^ft.  measure.  How  many 
times  did  you  apply  the  measure  ?     |^  ft.  -h  i  ft.  =  3. 

3.  Measure  ^  ft.  with  ^ft.  measure.  How  many 
times  did  you  apply  the  measure  ?     |^  ft.  ^  J  ft.  =  2. 

4.  Measure  ^  ft.  with  ^ft.  measure.  The  measure 
is  applied  ^  times.     -J-  ft.  h-  i  ft.  =  ^. 

5.  If  a  measure  ^  ft.  long  is  used  to  measure  \  ft., 
the  measure  would  be  applied  ^  times.    J  ft.  h-  i  ft.  =  ^. 

6.  If  a  measure  J  ft.  long  is  used  to  measure  ^  ft., 
would  the  measure  be  applied  more  or  less  than  1 
time  ?  With  your  measure  find  what  part  of  |  ft.  is 
used  to  measure  J  ft.     ^  f t.  -j-  i  ft.  =  — . 

7.  Measure  f  ft.  by  f  ft.  by  changing  both  to  12ths : 
f  ft.  =  3^  ft. ;  f  ft.  =  3%  ft.     Measure  j%  ft.  by  ^  ft. 

8.  Divide  f  by  f .  This  may  be  done  by  changing 
the  fractions  to  a  common  denominator  and  dividing 
the  numerators,  thus  :  f-^|  =  x2"^T^'^^"^^~-'^8"- 

336.  The  following  is  a  shorter  method : 

7b  divide  a  fraction  hjj  a  fraction,  invert  the  divisor 
and  midtiphj. 

1.    Divide  |  by  f . 

Model:  f-f  =  f  x  f  =  f,  or  1|. 


2.   i-f 
3-    f-f 


A-l 

14  _^  7 
W^8 

H- 

-f 

\i-i 

n-f 

1  r>  .  8 
TT--J7 

1  '.\ 
T4" 

-h 

H-l 

DIVISION   OF   FRACTIONS  221 

337.   Written  Exercises. 

a.    Divide  6|  by  |. 

Model:  61  =  -^^.     |>^i  =  f  x5  =  |,  or  Sf 

Solve : 


a 

b 

C 

rt 

e 

2. 

6|^| 

121- 

-f 

74    .      3 
'5  ^    5 

lOf-f 

9f-l 

3. 

8|-| 

18|- 

-i 

yl    .      2 
^6-"    3 

45f^f 

lU-i 

4. 

n^i 

25f 

-f 

9f-| 

7|-i 

8^-1 

5. 

7f-| 

12|- 

.^3 
•    4 

9f-A 

8i-| 

H^h 

338.   Written  Exercises. 

1.    Divide  4|  by  3f . 

Model  c  ^ 

42=14.    35  =  2_3_        14^28^14       1^28    ^j.  1  _5_ 

Divide : 


a 

6 

C 

eJ 

2|  by  4f 

8i  by  2f 

5J  by  4f 

4|.  by  3| 

6*  by  21 

9|  by  31 

9iby8i 

2|  by  3| 

n  by  6| 

5|  by  7| 

81  by  6f 

91  by  41 

5.    5|by8f         4fby3f         5|  by  3^         6|  by  4i 
339.    Written  Exercises. 


a 

b 

C 

d 

1. 

16f-12f 

53|  +  34f 

8|x    4t 

651^ 

i 

2. 

40f  +  17f 

87|-70f 

9|xl0 

241 X 

5f 

3. 

56f-25l 

24f  +  14f 

65|^   4 

401- 

34| 

4. 

6.5f  +  37f 

44*  -  14f 

3f-   41 

5f- 

C| 

5. 

18|x    6f 

451  +  541 

6-1-17 

19f- 

9f 

6. 

37|^   41 

691 _  43| 

56§^   8 

4|x 

4f 

222  FRACTIONS   AND   DECIMALS 

340.  Written  Exercises. 

1.  Find  tiie  surface  of  a  walk  12  ft.  8  in.  (12|  ft.) 
long  and  6  ft.  9  in.  (6|  ft.)  wide. 

2.  Find  the  cost  of  laying  cement  on  the  same  walk 
at  $  .12 J  per  square  foot. 

3.  One  field  contains  36f  A. ;  another  22|  A.  Find 
the  number  of  acres  in  the  two  fields. 

4.  If  a  train  travels  560-|  mi.  in  12  hr.,  what  is 
the  average  rate  per  hour  ? 

'  5.    If  it  cost  $  36,000  to  build  3^  mi.  of  trolly  line, 
how  much  on  the  average  did  it  cost  to  build  1  mi.? 

6.  What  will  be  the  cost  of  16  doz.  8  eggs  (16|  doz.) 
at  16^  a  dozen? 

7.  A  man  rented  his  farm  for  ^  of  the  crop.  He 
received  for  his  share  of  oats  360  sacks.  How  many 
sacks  of  oats  were  raised  on  the  farm  ? 

341.  Oral  Exercises. 

1.  What  part  of  1  doz.  eggs  are  8  eggs  ? 

2.  Express  in  dozens  and  fraction  of  a  dozen  :  3  doz. 
5 ;  4  doz.  8 ;  6  doz.  7. 

3.  Express  in  feet  and  fraction  of  feet :  3  ft.  6  in.; 
4  ft.  8  in.;  9  ft.  7  in. 

4.  Find  the  cost  of  5  doz.  6  eggs  at  16^^  per  dozen. 

5.  When  coal  is  selling  at  $  8  per  ton,  what  part 
of  a  ton  can  be  bought  for  $  2  ?    For  $  4  ?    For  $  6  ? 

6.  How  many  pieces  of  string  each  |  ft.  long  can 
be  cut  from  a  string  6  ft.  long  ? 


WRITTEN   EXERCISES  223 

342.    Written  Exercises. 

1.  A  farmer  raised  9|  tons  of  hay  on  4  acres. 
What  was  the  yield  per  acre  ? 

2.  A  man  bought  a  house  and  a  50-ft.  lot  for 
$  4500.  If  the  house  was  valued  at  $  3500,  what 
was  the  value  of  the  land  per  front  foot  ? 

3.  When  cloth  is  selling  at  f  dollar  per  yard, 
how  many  yards  can  be  purchased  for  $  3  ? 

4.  A  merchant  bought  cloth  at  $  .621  a  yard. 
He  sold  it  at  $  .75  a  yard.  How  much  did  he  gain 
on  each  yard  ? 

5.  A  merchant  bought  cloth  at  $  .37^^  a  yard,  and 
sold  it  at  $  .25  a  yard.  How  much  did  he  lose  on 
each  yard  ? 

6.  When  hay  is  selling  at  $  8  a  ton,  how  many 
tons  can  be  purchased  for  $  12  ?    $6?    $9?    $15? 

7.  Divide  7f  in.  by  2 ;  8f  ft.  by  3  ;   6|  A.  by  2  ; 

T%  in-  by  |. 

8.  A  woodchopper  cut  12  cords  of  wood  for  $  18. 
How  much  was  that  a  cord  ? 

9.  A  boy  has  25^  in  nickels,  30^  in  dimes,  $3 
in  quarter-dollars,  and  $  5  in  half-dollars.  How  many 
pieces  of  money  has  he  ? 

10.  One  square  inch  is  what  part  of  a  square  foot? 

11.  One  square  foot  is  what  part  of  a  square  yard? 

12.  Divide  f  by  2;    f  by  4  ;   f  by  f  ;  f  by  5  ;   12| 
by  6;    2fby5;    6|  by  4. 


224  FRACTIONS  AND   DECIMALS 

RATIO    AND    PROPORTION 

343.   1.    12  apples  are  worth  —  times  as  much  as 
6  apples. 

2.  4  apples  are  —  third  of  12  apples. 

3.  12  apples  are  worth  —  times  as  much  as  4 
apples. 

4.  If  4  apples  are  worth  5^,  12  apples  are  worth 
—  ^. 

5.  If  8  apples  are  worth  10^',  24  apples  are  w^orth 
— ^. 

6.  If  7  bu.  of  corn  are  worth  $  4.20,  21  bu.  are 
worth  $  — . 

7.  If  4  lb.  of  coffee  are  worth  $1,  6  lb.  of  coffee 
are  worth  $  — . 

a  If  21  yd.  of  cloth  cost  $  2,  5  yd.  of  it  will 
cost  $ . 

9.  121  is  _  half  of  25;  —  eighth  of  100;  — 
fourth  of  50;  — sixth  of  75;  — third  of  37J-;  — 
fifth  of  621. 

10.  If  25  yards  of  cloth  cost  $2.12,  at  the  same 
rate  how  much  will  50  yd.  of  the  cloth  cost  ? 

11.  331  is of  100.      75  is of  100. 

66|  is of  100.     371  is of  100. 

12.  66f  bu.  of  wheat  are  w^orth  —  thirds  as  much 
as  100  bu. 

13.  371  bu.  of  wheat  are  worth  —  eighths  as  much 
as  100  bu. 


RATIO   AND   PROPORTION  225 

14.  10  brooms  are  worth  —  times  as  much  as  4 
brooms  of  the  same.  kind. 

15.  1  of  6  is  i-  of  — .     1  of  8  is  i  of  — . 

16.  f  of  9  apples  are  |  of  —  apples. 

17.  t  of  28  ft.  are  —  ft.  8  ft.  are  |  of  —  ft. 

18.  If  5  tons  of  hay  will  feed  6  horses  a  certain 
time,  15  tons  will  feed  —  horses  for  the  same  time. 

19.  Twelve  sacks  of  barley  will  feed  6  horses  as 
long  as  —  sacks  will  feed  24  horses. 

20.  Working  for  the  same  wages,  5  men  can  earn 
as  much  in  12  weeks  as  10  men  can  earn  in  —  weeks. 

21.  The  ratio  of  5  men  to  10  men- is  the  same  as 
the  ratio  of  12  bu.  to  —  bu. 

22.  The  ratio  of  |^  to  |  is  the  same  as  the  ratio  of 
f  to— . 

Find  the  cost : 

23.  Of  16  apples  if  8  apples  cost  12^. 

24.  Of  32  sacks  of  barley  if  8  sacks  cost  $  7.     . 

25.  Of  25  tons  of  hay  if  100  tons  cost  $  824. 

26.  Of  331  yd.  of  cloth  if  66f  yd.  cost$  36. 

27.  Of  121  boxes  of  apples  if  37^  boxes  cost  $  27. 

28.  Of  10  tons  of  grapes  if  4  tons  cost  $  48. 

29.  Of  8  sheep  if  24  sheep  cost  $  45. 

30.  Of  11  yd.  of  cloth  if  44  yd.  cost  $  22. 

31.  Of  1  sq.  yd.  of  blackboard  if  1  sq.  ft.  cost  20^. 

32.  Of  IJ  doz.  chickens  if  3  doz.  cost  $  12. 

1st  Bk  Ahith — 1.5 


226  FRACTIONS  AND  DECIMALS 

AREAS 

344.  1.  A  rectangle  5  ft.  long  and  2^  in.  wide  con- 
tains —  sq.  ft. 

2.  A  schoolroom  is  30  ft.  long  and  26  ft.  wide. 
The  floor  contains  —  sq.  ft. 

3.  How  many  corners  are  there  on  a  cube  ?  On 
a  box?  How  many. faces  are  there  on  a  cube?  On 
a  box? 

4.  Find  the  area  of  one  face  of  a  3-in.  cube. 

5.  Measure  the  length,  width,  and  height  of  any 
box.  Using  these  dimensions,  find  the  following :  the 
area  of  one  of  the  ends ;  the  area  of  the  two  ends ; 
the  area  of  one  of  the  sides  ;  the  area  of  the  two  sides ; 
the  area  of  the  bottom  of  the  box ;  the  area  of  the 
top  and  bottom ;  the  area  of  the  six  faces. 

6.  Is  the  room  you  are  in  shaped  like  a  box  ? 

7.  How  many  faces  has  the  room  ?  The  floor  of 
the  room  corresponds  to  the  -^  of  the  box  ;  the  ceiling 
of  the  room  to  the  —  of  the  box ;  the  sides  of  the 
room  to  the  —  of  the  box ;  and  the  end  of  the  room  to 
the  —  of  the  box. 

8.  The  man  who  plastered  your  schoolroom  was 
paid  a  price  per  square  yard  for  doing  the  work.  If 
he  received  21^  per  square  yard.,  how  much  did  he 
receive  for  plastering  the  room  ? 

9.  Think  of  a  room  that  is  15  ft.  long,  12  ft.  wide, 
and  10  ft.  high.     Find  : 

a.   The  area  of  the  floor ;    h.    the  area  of  the  sides. 


DECIMAL  FRACTIONS  227 

DECIMAL  FRACTIONS 

345.  1.  Divide  27  by  4.  Divide  32  by  5.  Read 
the  answers. 

2.  Divide  27  by  10.  Divide  32  by  10.  Read 
the  answers. 

3.  2^  is  also  written  2.7.     S^q  is  also  written  3.2. 

4.  The  period  between  2  and  7  in  2.7  is  called 
the  decimal  point.  The  fraction  ^V  may  be  written  .7. 
Any  fraction  whose  denominator  is  10  may  be  so 
written.  The  form  j-V  ^^  ^^^  form  of  a  common 
fraction.    The  form  .7  is  the  form  of  a  decimal  fraction. 

5.  7|  is  the  quotient  of  37 -f- 5.  Is  the  divisor 
found  in  the  quotient  ?     If  so,  where  ? 

6.  Does  the  divisor  appear  in  the  quotient  of  27  -^ 
10?     Where? 

7.  3|-  is  the  quotient.  Find  the  divisor,  the  re- 
mainder, and  the  dividend. 

8.  The  following  are  quotients.  Name  the  divisors, 
the  remainders,  and  read  the  quotients :  8|,  Q^,  5^^ 
6.4,  5i|o,  6.7,  50.52,  6.75,  9.38. 

9.  Change  the  following  to  decimals:  8^,  9^^^, 
18 J^,  27,-%,  7,^,  V,. 

10.  Read  the  following:  8.1,  17.5,  20.8,  9.9. 

11.  Write  as  common  fractions:  .6,  .9,  .5,  .4. 

12.  Divide  the  following  by  10  by  placing  the  deci- 
mal point  between  the  units'  and  tens'  places  in  each: 
93,  84,  935,  61,  80,  400,  405.     Read  the  quotients. 


228  FRACTIONS   AND   DECIMALS 

346.  1.  .7  is  read  — ;  .07  is  read  — ;  .007  is  read 
seven  thousandths. 

2.  Ten  cents  is  what  part  of  one  dollar  ?     Write 
ten  cents,  using  the  dollar  sign. 

3.  Write- one  cent,  using  the  dollar  sign. 

4.  $.10  is  how  many  times   $.01?     .10  is   how 
many  times  .01  ? 

5.  Which  is  more,  .8  mi.,  .08  mi.,  or  .008  mi.? 

6.  Which  is  more  .6  mi.,  .60  mi.,  or  .600  mi.  ? 

7.  Compare  :  .4,  .40,  .400.    Compare  :  .4,  .04,  .004. 

^      10  — T0¥—  1000*       To  FO  —  TF(J  —  TO  • 

9.   Express  the  fractions  in  Exercise  8  in  the  form 
of  decimal  fractions. 

10.  A  decimal  fraction  is  one  whose  denominator  is 
10  or  some  power  of  10,  as  100,  1000,  10,000,  etc. 

11.  Read:  .8,  .67,  .672.  When  the  number  of 
places  to  the  right  of  the  decimal  point  is  one,  the 
denominator  is  — ;  when  the  number  of  places  is  two, 
the  denominator  is  — ;  when  the  number  of  places  is 
three,  the  denominator  is  — . 

12.  In  the  decimal  .7  what  is  the  numerator? 
What  is  the  denominator  ? 

13.  What  is  the  numerator  of  the  fraction  .05? 
Of  .324? 

14.  In  .72  what  is  the  numerator  ? 


NOTATION   AND  NUMERATION   OF   DECIMALS       229 

NOTATION   AND    NUMERATION   OF   DECIMALS 

347.  1.  The  names  of  the  orders  to  the  right  of 
the  decimal  point  are  : 

First :      Tenths'  order  .     .     ; 8 

Second  :  Hundredths'  order 67 

Third:     Thousandths' order .672 

Fourth :  Ten-thousandths'  order     .     .     .     .6789 

Fifth :      Hundred-thousandths'  order  .     .     .67898 

-  Sixth  :      Millionths'  order 678968 

2.  Memorize  the  above.  Remember  that  four  deci- 
mal places  give  ten-thousandths ;  that  five  decimal 
places  give  hundred-thousandths,  etc. 

2h  read  a  decimal,  read  the  number  ivithout  refer- 
ence to  the  decimal  point,  and  then  add  the  name  of 
the  order  of  the  right-hand  figure  of  the  numerator. 

3.  .62  is  read  sixty-two  hundredths.  It  is  given 
the  name  of  the  second  order,  hundredths. 

4.  .00062  is  read  sixty-two  hundred-thousandths. 
Read  .0062. 

5.  6.72  is  read  six  and  seventy-two  hundredths. 
Read :  24.52,  5.672,  6.08,  52.004,  .52  oz.,  6.5  oz. 

6.  Write  in  a  column,  with  the  decimal  points 
directly  below  one  another,  and  read :  .272,  .27,  7.62, 
7.3,  4.67,  9.787,  6.72896. 

7.  Write  in  a  column  :  four  and  sixty-two  hun- 
dredths, five  and  sixty-five  thousandths,  seven  and 
six  tenths. 


230  FRACTIONS   AND  DECIMALS 

ADDITION  OF  DECIMALS 

348.  1.    Find  the  sum  of  1.27,  36.2,  and  54.036. 

Write  the  numbers  so  that  the 

Model  :  1.27       decimal  points  are  directly  below 

36.2         one   another.     Add    as   in  whole 

54.036     numbers.    Place  the  decimal  point 

91.506     in    the   sum   directly    below    the 

decimal  point  in  the  addends. 

2.  Add:    36.5,42.47,  62.367,48. 

3.  Add:    7.27,52.005,64.3,52. 

4.  Add  :  .05,  1.0501,  10.504,  150.41,  .546. 

SUBTRACTION  OF  DECIMALS 

349.  1.    From  5.2  subtract  2.27. 

Consider  5.2  as  5.20.      Subtract 
Model:  5.2      as   in   whole   numbers.     Place   the 
2.27    decimal  point  in  the  answer  directly 
2.93    below  the  decimal  point  in  the  sub- 
trahend. 

2.  From  16.7  subtract  10.25. 

3.  From  126  subtract  8.75. 

4.  How  much  more  is  35.12  than  14.6? 

5.  How  much  less  is  84.7  than  125.5  ? 

6.  A  man  owned  127.7  A.  of  land.  He  sold  27.9  A. 
to  one  neighbor  and  30.5  A.  to  another.  How  many 
acres  had  he  left  ? 


FRACTIONS   CHANGED   TO   HUNDREDTHS         231 

FRACTIONS  CHANGED  TO  HUNDREDTHS 
350.   1.    Memorize  the  following : 

2^   ~"TFO  "~  •^^*  4~10  0~~-^'^-  25  ""  10  0  ~  •^^^• 

JL   _    2  5    —    25  3.__75_—    75  2    —    20    —    OQ 

t~Too  — •^^-  4— loo--'^'  10  — 100  — •^^• 

1    —    20—90  2  —    40    _    40  3    _    30    —    QO 

5"-roo--^^-  :5^- 100 -•^'^-  To-Ton^--^^- 

1_10_10  3_60_A0  4_40_i0 

TO-TOO--'^^-  5-T^O--'^^-  TO-T¥^--^^- 

1    _      5      _    05  4_    80    _    CO  1    _      2      _    (W 

20  -  TOO  -  '^^'  5  -  TOO  -    O^-  50  -  Too  "  •^^^• 

2.  Express  as  common  fractions  :  .80,  .75,  .25,  .50, 
.60,  .40,  .30,  .20. 

3.  Express  as  common  fractions  :  .05,  .02,  .04,  .01, 
.10,  .70,  .80,  .90. 

4.  Write  with  the  fractional  part  expressed  as  a 
decimal:  6|,  71,  8f,  12^  25 J^,  28^,  ^,  3f,  14f, 
221 

.5.  Write  with  the  decimal  part  expressed  as  a 
common  fraction:  4.25,  6.20,  7.75,  12.80,  15.04, 
35.02,  27.75,  15.60. 

6.  Write  as  decimal  fractions  and  add :  f ,  \^  3^, 

_3_    3    JL 
20'   5'  50* 

7.  Change  to  common  fractions  and  reduce  to 
lowest  terms :    .375,  .125,  .625,  .875. 

8.  What  effect  upon  the  value  of  .25  has  annexing 
a  cipher  to  the  right  of  it,  thus :  .250  ? 

9.  What  effect  upon  the  value  of  .25  has  the  plac- 
ing of  a  cipher  before  it,  thus :  .025  ? 

10.    Express  in  dollars  and  cents:   $6|,  $8 J,  $9 J, 

$7-g-,    $lU-j_0-,    $1^4,    ^IOyq-,    $0^-q,    $7;g^-Q. 


232  FRACTIONS   AND   DECIMALS 

MULTIPLICATION  OF  DECIMALS 

351.    1.    Read  the  following :  5,  .5,  .05,  and  .005. 

2.  Compare  the  value  of  5  and  .5  ;  of  .5  and  .05. 

3.  Compare :    625  ft.  and  62.5  ft. ;  62.5  ft.  and 
6.25  ft. ;  6.25  ft.  and  .625  ft. 

4.  Moving  the  decimal  point  one  place  to  the  left 
has  what  effect  upon  the  value  of  a  number  ? 

5.  Compare:    .385  ft.  and  3.85  ft.;   3.85  ft.  and 
38.5  ft. ;  38.5  ft.  and  385  ft. ;  385  ft.  and  3850  ft. 

6.  Moving  the  decimal    point  one   place  to  the 
right  has  what  effect  upon  the  value  of  a  quantity  ? 

7.  What  part  of  22  is  2.2  ?   Of  30  is  3  ?    Of  3  is  .3  ? 

8.  Multiply:  62  by  10;  36  by  10;  675  by  10. 

9.  Multiply:  ^  by  10 ;  ^  by  10 ;  ^r^  by  10. 

10.  Multiply  :  .3  by  10  ;  .03  by  10 ;  .003  by  10. 

11.  Multiply  in  the  shortest  way  possible :  2  by 
10  ;  .2  by  10  ;  .4  by  10  ;  37  by  10  ;  3.7  by  10  ;  .37 
by  10. 

12.  Multiply  each  by  10  :  .67,  5.2,  .52,  6.27,  7.89. 

13.  Compare  the  value  of  $1.84  and  $184;  of 
$125  and  $1.25. 

'i4.    Multiply   each   by    100:    $6.25,  $.50,  $.05, 
$1.05. 

15.  Multiply  12  by  ^V-     Multiply  12  by  .1. 

16.  Divide  12  by  10.     12  x  .1  =  1.2. 

17.  Multiplying  a  number  by  -^  is  the  same  as 
dividing  the  number  by  — . 


MULTIPLICATION  OF  DECIMALS  233 

352.  To  multiply  a  decimal  by  an  integer,  multiply 
as  in  whole  numbers.  Point  off  in  the  answer  as 
many  decimal  places  as  there  are  decimal  places  in 
the  midtiplicand. 

1.    Multiply  42.35  by  8.         Model  :  42.35 


338.80 
2.   63.75x5         3.   327.42x9         4.   6.843x105 

353.  To  multiply  an  integer  by  a  decimal,  multiply 
as  in  ivhole  numbers.  Point  off  in  the  answer  as 
many  decimal  places  as  there  are  decimal  places  in 
the  multiplier. 

1.    Multiply  367  by  8.2.  Model  :     367 

8.2 
734 
2936 


3009.4 
2.    325x8.4  3.    863  X. 94  4.    754  x  .38 

354.  To  midtiply  a  decimal  by  a  decimal,  multiply 
as  in  ivhole  numbers.  Point  off  in  the  answer  as  many 
decimal  places  as  there  are  decimal  places  in  both  mul- 
tiplier and  midtiplicand. 

1.  Multiply  5.25  by  4.7.        Model  :     5.25 

_M 

3675 
2100 
24.675 

2.  6.24x4.7  3.   95.7  X. 56  4.    .875x4.5 


234  FRACTIONS  AND  DECIMALS 

355.  Written  Exercises. 

1.  6.34x2.4  5.  .937x6.3  9.  .036x4.5 

2.  83.6  x. 36  6.  .372  X. 27  10.  .004x4.05 

3.  745  X. 67  7.  67.5  X. 04  11  .405  x  .006 

4.  8.32x45  8.  9.67  X. 003  12.  5.07  x  .001 

PERCENTAGE 

356.  1.  Read :  Y^Q,  .04.  This  may  be  written  thus: 
4%.  It  is  then  read,  four  per  cent.  Per  cent  means 
hundredths. 

2.  5%  is  the  same  as  -^q,  or  .05. 

3.  Express  as  per  cent:  ^f^,  ^,  ^,  ^^^,  /^o^. 

4.  Express  as  per  cent:  .03,  .09,  .12,  .40,  .75,  .01. 

5.  Express  as  fractions:  11%,  15%,  20%,  35%,  80%. 

6.  Express  as  decimals:  4%,  18%,  7%,  75%,  10%. 

7.  6%  of  $65  is  the  same  as  $65  x  .06. 

8.  Find  8%  of  $500;  of  $250;  of  $1000. 

9.  Find  7%  of  $100;  of  $62.50;  of  $83.75. 

10.  To  find  10%  of  any  number,  divide  the  number 
by-. 

11.  Find  10%of  $250;  of  $340;  of  $400;  of  $1000. 

12.  25%  is  the  same  as  ^.  To  find  25%  of  a  num- 
ber, divide  the  number  by  — . 

13.  Find  25%  of  $40;  of  $800;  of  $1200;  of  $4. 

14.  50%  is  the  same  as  — .  To  find  50%  of  a  num- 
ber, divide  the  number  by  — . 

15.  Find50%of  $80;  of  $100;  of  $400;  of  $1000. 


FRACTIONS  AS  PER  CENTS  235 

FRACTIONS   AS  PER  CENTS 

357.  To  change  a  fraction  to  per  cent,  multiply  the 
fraction  by  100.  Reduce  the  product  to  a  whole  or  a 
mixed  number. 

1.    Change  |  to  per  cent. 


Model:  f  x  100  =  ^f^  =  40.     f  =  ^u7^. 

Change  to  per  cent :  J,  |,  |,  |,  ^,  },  | 

2.    Memorize  the  folloioing : 

1=50%.  i=16t%.  |=66|%. 

1=33^%.  i=14f%.  1=75%. 

1  =  25%.  i=12i%.  1=371%. 

1  =  20%.  ^0  =  10%.  1  =  621%. 

358.   Oral  Exercises. 

Change  the  per  cent  to  a  fraction  and  find  :  * 

1.  25%  of  200  ft.  7.    50%  of  800  mi: 

2.  20%  of  $150.  8.    25%  of  360  A. 

3.  331%  of  $210.  9.    20%  of  100  yd. 

4.  75%  of  400  ft.  10.    14f  %  of  70  yr. 

5.  66f%  of  $120.  11.    121%  of  $720. 

6.  371%  of  $80.  12.   10%  of  $950. 

13.  What  is  20%  of  $375?  10%  of  $236?  50%  of 
$278?  10%  of  362  gal.?  25%  of  640  lb.?  75%  of 
360  A.? 

14.  What  is  331%  of  360  A.?  66f%of  $120?  14f% 
of  $140?   16|%  of  $180?  371%  of  $800? 

*  Supplement  this  exercise  with  oral  drill  until  the  pupils  are  able  to 
find  the  above  per  cents  readily  by  the  use  of  their  fractional  equivalents. 
The  fractional  equivalents  of  the  above  per  cents  should  be  used  in  subse- 
quent exercises. 


236  FRACTIONS   AND  DECIMALS 

359.  Written  Exercises. 

1.  Merchandise  is  generally  sold  at  a  certain  per 
cent  profit  on  the  cost.  What  per  cent  of  profit  do 
you  think  a  grocer  should  make  on  tea  ?  *  On  sugar  ? 
On  strawberries? 

2.  If  a  grocer  buys  tea  at  30^  a  pound,  and  sells  it  at 
a  profit  of  33 J%,  what  is  the  selling  price  per  pound? 

3.  A  merchant  bought  a  suit  for  $15  and  sold  it 
at  a  profit  of  20%.     What  was  his  profit  ? 

4.  Locate  Ogden,  Omaha,  and  San  Francisco  on 
the  map.  It  is  844.7  mi.  from  Ogden  to  San  Fran- 
cisco, and  1004.7  mi.  from  Ogden  to  Omaha.  How 
far  is  it  from  San  Francisco  to  Omaha  ?  How  much 
nearer  is  Ogden  to  San  Francisco  than  to  Omaha  ? 

5.  A  merchant  bought  cloth  at  $.12  a  yard,  and 
sold  it  at  a  gain  of  25%.  What  was  his  profit  on 
each  yard  ?     What  was  the  selling  price  per  yard  ? 

6.  A  clothing  store  advertised  boys'  suits  worth 
$12  at  25%  reduction.  Find  the  amount  of  the 
reduction  and  the  cost  of  a  suit. 

7.  A  dry  goods  store  advertised  a  20%  reduction 
sale  on  carpets.  Find  the  reduction  per  yard  on  car- 
pets that  formerly  sold  at  60^  a  yard. 

8.  A  dealer  in  farm  implements  bought  carriages 
at  $50  and  sold  them  at  a  profit  of  20%.  Find  his 
profit  on  each  carriage  sold. 

*  Discuss  these  and  similar  questions  with  class. 


INTEREST  237 


INTEREST 


360.     1.    When   one  rents  a  house  from  another, 
how  does  he  usually  pay  for  its  use  ? 

2.  When  one  rents  a  farm  from  another,  how  does 
he  usually  pay  for  its  use  ? 

3.  When  one  borrows  money  from  another,  how 
does  he  usually  pay  for  its  use  ?  What  is  the  name 
given  to  money  paid  for  the  use  of  money  ? 

4.  What  is  the  meaning  of  the  following  :  ''  Money 
to  loan  on  good  securities.     Interest  5%."  ? 

5.  A  man  borrowed  $600  for  one  year.  He  paid 
f  .05  for  the  use  of  each  dollar,  or  5%  of  the  amount 
borrowed.     How  much  interest  did  he  pay  ? 

6.  A  boy  borrowed  $40  from  his  father  to  buy  a 
bicycle.  He  agreed  to  pay  his  father  5%  interest. 
How  much  interest  should  he  pay  each  year  ? 

7.  A  contractor  borrowed  $3000  at  6%  and  used 
the  money  to  build  a  house,  which  he  rented  at  $20 
a  month.  Find  the  interest  which  he  must  pay  each 
year.  Find  the  amount  of  rent  which  he  receives 
each  year.  How  much  more  does  the  rent  amount  to 
than  the  interest  ? 

a    Find  the  interest  on  $1800  for  1  year  at  6%. 

9.    Find  the  interest  on  $2000  for  1  year  at  8%. 

10.  Find  the  interest  on  $2000  for  2  years  at  8%. 

11.  Find  the  interest  on  $6500  for  1  year  at  ^%. 
The  money  borrowed  or  loaned  is  called  the  principal. 


238  FRACTIONS  AND  DECIMALS 

DIVISION  OF   DECIMALS 

361.  Oral  Exercises. 

1.  4  pt. -f- 2  pt.  =  — .     8qt.^2qt.  =  — . 

2.  16  gal.  -5-  2  gal.  =  — .     10  mi.  -f-  2  mi.  =  — . 

3.  8  tenths -^2  tenths  =  — .  4  hundredths -^  2 
hundredths  =  — . 

4.  .8^.2  =  —.     ,04-^.02  =  —. 

5.  .12^.06  =  —.     .8-^.4  =  —. 

6.  .008-^.002  =  —.     .044-^.004  =  —. 

7.  If  the  divisor  contains  tenths,  tenths  of  the 
dividend  may  give  a  whole  number  in  the  quotient. 
1.6  ft. -^  .2  ft.  =  8. 

8.  If  the  divisor  contains  hundredths,  hundredths 
of  the  dividend  may  give  a  whole  number  in  the  quo- 
tient.    .16  ft. -^.02  ft.  =  8. 

362.  Written  Exercises. 
1.   Divide  12.2  by  .2. 

61. 
Model:  .2)12.2  Note  first  the  lowest  order  in 
the  divisor.  In  this  case  tenths  is  the  lowest  order 
in  the  divisor.  Place  the  decimal  point  above  and 
after  the  figure  of  the  dividend  occupying  tenths' 
place.  Divide  without  reference  to  the  decimal  point 
in  the  quotient. 

Place  the  decimal  point  in  the  quotient  above  and 
after  the  figure  in  the  dividend  occupying  the  same 
order  or  place  as  the  lowest  order  in  the  divisor. 


DIVISION  OF  DECIMALS  289 

2.   Divide  .66  by  .02 ;  .328  by  .04. 


Model  :  .02).66     Model  :  .04).328 

3.  Divide  6.6  by  .03. 

220. 
Model:    .03)6.60     As   the  divisor  contains   hun- 
dredths, change  the  dividend  to  hundredths  by  an- 
nexing a  cipher.     Place  the  decimal  point  above  and 
after  hundredths'  place.     Why  ? 

4.  Divide  12  by  .002. 
Model:  .002)12.000' 

5.  Divide  .126  by  2. 

.063 
Model:  2.). 126  As  units'  order  is  the  lowest 
order  in  the  divisor,  place  the  decimal  point  above 
and  after  the  units'  place  of  the  dividend.  2  is 
contained  in  1  no  times.  Write  0  in  the  quotient. 
Complete  the  division. 

363.    Arrange  as  in  the  models  and  fix  the  decimal 
point  in  the  quotient : 


a 

6 

c 

d 

1. 

1.26^4 

.36^.02 

3.6^2 

360 ->  .002 

2. 

.52-^.2 

72.8^6 

7.2^2 

360^7.5 

3. 

1.55  H- 5 

20.65^5 

15.5^.05 

4.2^32.62 

4. 

1.6^2 

16  H- .002 

.16^20 

.678-^629 

5. 

2.4 -^  6 

.240^.006 

2.40^12 

.54^.0008 

240  FRACTIONS   AND  DECIMALS 

364.  Written  Exercises. 

Arrange  as  in  the  models  and  fix  the  decimal  point 
before  dividing  : 

1.  Divide  12  by  .02;  82  by  .04;  5.2  by  .02. 

2.  Divide  3.6  by  .2;  7.2  by  .2;  .72  by  .02. 

3.  Divide  13.6  by  .02 ;  6.66  by  .002. 

4.  Divide  .155  by  5 ;  .2065  by  5. 

5.  Divide  each  by  .002  :  1.6,  16,  .16,  .0016. 

6.  Divide  each  by  5 :  1.5,  .015,  .15,  .0015. 

7.  Divide  each  by  .05  :  .35,  3.5,  .035,  .003.5. 
^8.   Divide  each  by  .025 :  62.5,  625,  6.25,  .0625. 
'  9.    Divide  each  in  No.  8  by  2.5  ;  by  25dr 

^  10.   Divide  each  by  9.3 :  23.25,  4.65,  465,  .0465. 
IT.    Divide  each  by  $'.96:  $240,  $2.40,  $.48. 

12.  Divide  $170  by  $.85  ;  138  by  95 ;  625  by  .25. 

13.  At  $.96  each,  how  many  books  can  be  bought 
for  $  2.40  ?     For  $  24  ?     For  $  4.80  ?     For  $  48  ? 

365.  Oral  Exercises. 

1.  Find  6%  of  $350.     Find  8^%  of  $250. 

2.  Find  71%  of  $360.     Find  9^%  of  $360. 

3.  Find  3%  of  $60;  of  $27;  of  $35;  of  $3.50. 

4.  Find  21%  of  $12;  of  $20;  of  $30;  of  $24. 

5.  Find  8%  of  $60;  of  $90;  of  $75;  of  $62. 

6.  3%  of  my  money  is  $  9.     Find  1  %  of  my  money. 

7.  1%  of  my  money  is  $3.     Find  100%  of  my 
money. 


DIVISION   OF   DECLMALS  241 

366.    Oral  Exercises. 

1.  7  is  1  of  14.     7  is— %  of  14. 

2.  6  is  —  third  of  18.     6  is  —  %  of  18. 

3.  5  is  —  third  of  15.     5  is  —  %  of  15. 

4.  7  is  —  third  of  21.     7  is  —  %  of  21. 

5.  1  pt.  is  —  half  of  1  quart.     1  pt.  is  —  %  of  1 
quart. 

6.  1  qt.  is  —  fourth  of  1  gaL     1  qt.  is  —  %  of  1 
gal. 

7.  1  ft.  is  —  third  of  1  yd.     1  ft.  is  —  %  of  1  yd. 

8.  3  in.  are  —  fourth  of  12  in.     3  in.  are  —  %  of 
12  in. 

9.  6  in.  are of  12  in.     6  in.  are  —  %  of  12  in. 

10.  2  pt.  are of  1  gaL     2  pt.  are  —  %  of  1  gal. 

11.  5  men  are of  15  men.     5  men  are  —  % 

of  15  men. 

12.  6  men  are  ^  of  —  men.     6  men  are  33  J  %  of 

—  men. 

13.  12  men  are  f  of  —  men.     12  men  are  40  %  of 

—  men.  • 

14.  $  5  is  1  of  $— .     $5  is  20%  of  $— . 

15.  16  men  are  ^  oi  —  men.     16  men  are  50  %  of 

—  men. 

16.  A  boy  sold  an  article  for  |  of  its  cost.  He 
gained  what  part  of  the  cost  ?  He  gained  5  cents. 
Find  the  cost. 

17.  5%  of  my  money  is  $25.  Find  1%  of  my 
money.  .  Find  100%  of  my  money. 

Iht  Rk  a  kith—  1(5 


242  FRACTIONS  AND   DECIMALS 

367.    Written  Exercises. 

1.   Divide  4.628  by  89. 

■  Model  :     .052         Place  the  decimal  point  in  the 
89)4.628     quotient  above  and  after  the  order 
4  45       in  the  dividend  that  corresponds 
178     to  the  lowest  order  in  the  divisor, 
178     — in  this  case,  units.     89  is  con- 
tained in  4  no  times.     This  shows 
that  there  is  no  whole  number  in  the  quotient.     Do 
not  write  the  0  in  the  quotient.    89  is  contained  in  46 
no  times.    Write  the  0  in  tenths'  place  in  the  quotient. 
462  will  contain  89.     Complete  the  division. 

2.  Divide  each  by  87:  20.01,  391.5,  7.743,  2.2815. 

3< Divide   each   by  7.9 :    1.6195,   24.174,   .32232, 
481.11. 

j4,  Divide  each  by  5.23 :  42.886,  .8368,  26^15. 

5.  A  boy  paid  $  .12  for  6  oranges.     Find  the  cost 
of  1  orange. 

6.  A  boy  paid  $  .06  for  12  apples.     Find  the  cost 
of  1  apple. 

7.  A  farmer  sold  220  boxes  of  apples  for  $121. 
Find  the  selling  price  per  box. 

8.  Find  what  per  cent  8  is  of  16 ;  12  is  of  48;  9  is 
of  72;  64  is  of  128. 

9.  Find  what  per  cent  45  is  of  50  ;   40  is  of  50  ; 
30  is  of  20. 


WRITTEN  EXERCISES  248 

368.  To  multiply  a  number  by  25,  oS^,  66|,  50, 
etc.,  multiply  the  number  by  100  and  take  such  a  part 
of  the  product  as  the  multiplier  is  o/"  100. 

1.  Multiply  624  by  25. 

V 

Model  :     15600         25  is  i  of  100.     Multiply  624 
4)62400     by  100,  and  take  J  of  the  product. 

2.  Multiply:  78  by  66f ;  69  by  331;  240  by  371; 
240  by  621 ;  480  by  75. 

3.  Multiply:  360  by  25;  1876  by  75;  1728  by 
121;    5280  by  331;    144  by  50. 

369.  To  divide  a  number  by  25,  331,  371.^  66 1^  75^ 
etc.,  divide  the  numher  by  100  and  midtiply  the  residt 
by  the  inverted  form  of  the  fraction  that  indicates  the 
part  the  multiplier  is  0/  100. 


Model  :  14.40         66f  is  |  of  100.     Divide  1440 


1.  Divide  1440  by  66f . 

61 

3     by  100,  and  multiply  the  quotient 
2)43.20     byf. 
21.60 

2.  Divide:  982  by  331;  by  75;  by  66|;  by  621; 
by  371 

3.  Divide:    1728  by  75;    5280  by  37i;   5760  by 
331 ;    640  by  121  •    6335  by  14f . 


244  FRACTIONS   AND  DECIMALS 

370.  To  divide  a  number  by  200,  300,  2000,  etc., 
point  off  as  tnany  places  from  the  rig  Jit  as  there  are 
ciphers  in  the  divisor,  and  divide  by  the  left-hand 
figure  of  the  divisor. 

1.  Divide  627  by  200. 

Model  :    3.135         Point  off  two  places.     Divide 
2)6.27     6.27  by  2. 

2.  Divide  768  by  3000. 

Model  :    .256         Point  off  three  places.     Divide 
3).768     .768  by  3. 

3.  Divide :  5758  by  2000 ;  8520  by  2000 ;  68,960 
by  5000. 

371.  To  find  25^,  33^^,  66f^,  etc.,  of  a  number, 
take  such  a  part  of  the  number  as  the  required  per 
cent  is  of  100/^. 

1.  Find  331^  of  $7521. 

Model  :      $2507         331^  is  J  of  100?^ 
3)$  7521         Take  1  of  $7521. 

2.  Find  371^  of  $88  ;  331?^  of  66  ;  62^^  of  $64  ; 
75^0  of  $160;  871^  of  $64. 

3.  A  man  bought  a  farm  for  $1200.  He  sold  it 
at  a  profit  of  33^ ^^  What  was  his  gain?  What 
was  his  selling  price  ? 

4.  $12  is  f  of  what  nmnber?  $12  is  ^^fo  more 
than  what  number  ? 


CHAPTER  VI 
DENOMINATE   NUMBERS 

372.  Denominate  units  of  measure  have  been  estab- 
lished in  accordance  with  law,  or  custom,  to  measure 
values,  weight,  time,  length,  surface,  capacity,  etc. 

UNITED    STATES    MONEY 

.     10  mills  (m.)  =  1  cent  (^) 

10  cents  =  1  dime  (d.) 

10  dimes         =  1  dollar  ($) 

10  dollars       =  1  eagle  (E.) 

How  many  dollars  are  there  in  a  double  eagle  ? 

Beginning  with  the  one  of  least  value,  name  the 
coins  that  are  in  circulation. 

Beginning  with  the  one  of  least  value,  name  the 
bills  that  are  in  circulation. 

373.  PAP^R  MEASURE 

24  sheets    =  1  quire 

20  quires    =  1  ream 

2  reams     =  1  bundle 

5  bundles  =  1  bale 

How  many  sheets  of  paper  are  there  in  J  quire? 
In  2  quires  ?     In  ^  quire  ? 

245 


246  DENOMINATE   NUMBERS 

374.  COUNTING 

12  units  =  1  dozen  (doz.) 

12  dozen  =  1  gross 

12  gross  =  1  great  gross 

20  units  =  1  score 

Name  something  that  is  sold  by  the  gross. 

375.  TIME   MEASURE 

60  seconds  (sec.)  =  1  minute  (min.) 

60  minutes  =  1  hour  (hr.) 

24  hours  =  1  day  (da.) 

7  days  =  1  week  (wk.) 

52  weeks  =  1  year  (yr.) 

365  days  =  1  year 

366  days  =  1  leap  year 
100  years  =  1  century. 

A  centennial  year  is  one  whose  number  is  divisible 
by  100.  Centennial  years  whose  numbers  are  divisi- 
ble by  400,  and  other  years  whose  numbers  are 
divisible  by  4,  are  leap  years. 

376.  LIQUID    MEASURE 

4  gills         =  1  pint  (pt.) 

2  pints        =  1  quart  (qt.) 

4  quarts      =  1  gallon  (gal.) 

31-|  gallons  =  1  barrel  (bbl.) 

2  barrels     =  1  hogshead  (hhd.) 


WRITTEN  PROBLEMS  247 

377.   Written  Problems. 

1.  Find  the  number  of  years,  months,  and  days 
from  January  3,  1873,  to  November  1,  1904. 

Model  :  1904  yr.  11  mo.  1  da.  November  1, 
1873  yr.  1  mo.  3  da.  1 904,  is  the  first 
31  yr.  9  mo.  28  da.  day  of  the  elev- 
enth month  in  the  year  1904.  January  3,  1873,  is 
the  third  day  of  the  first  month  in  the  year  1873. 
Write  these  dates  as  above,  and  subtract.  3  da. 
cannot  be  taken  from  1  da.,  so  add  30  da.  (one 
month)  to  1  da.  Subtract  3  da.  from  31  da.  As 
one  month  was  added  to  the  minuend,  add  one  month 
to  the  subtrahend.  This  changes  1  mo.  to  2  mo. 
Subtract  2  mo.  from  11  mo.  Then  subtract  1873 
yr.  from  1904  yr.  The  answer  is  31  yr.,  9  mo.,  and 
28  da. 

2.  Abraham  Lincoln  w^as  born  February  12,  1809. 
He  died  April  15, 1865.    How  old  was  he  when  he  died? 

3.  George  Washington  died  December  14,  1799. 
How  many  years  is  it  since  his  death  ? 

4.  Daniel  Webster  died  October  24,  1852,  at  the 
age  of  70  yr.  9  mo.  6  da.   What  was  the  date  of  his  birth  ? 

5.  William  Penn  was  bom  October  14,  1644.  He 
died  at  the  age  of  73  yr.  9  mo.  16  da.  What  was  the 
date  of  his  death  ? 

6.  A  man  borrowed  money  January  16,  1902. 
He  paid  it  March  7,  1903.  How  long  did  he  keep 
the  money? 


248  DENOMINATE  NUMBERS 

378.  AVOIRDUPOIS    WEIGHT 

16  ounces  (oz.)  =  1  pound  (lb.) 
100  pounds         =  1  hundredweight  (cwt.) 
2000  pounds         =  1  ton  (T.) 

The  long  ton  contains  2240  pounds.  It  is  used  at 
the  custom-houses  in  invoices  of  some  imports,  and 
sometimes  in  weighing  coal. 

379.  Oral  Exercises. 

1.  Hold  up  enough  books  to  weigh  about  one  pound. 

2.  An  ounce  is  what  part  of  a  pound  ? 

3.  How  many  hundredweight  are  there  in  600  lb.  ? 
In  624  lb.  ?     In  52  lb.  ?     In  167  lb.  ? 

4.  How  many  pounds  are  there  in  ^  cwt.  ?  In  2^ 
cwt.  ?     In  3f  cwt.  ?     In  IT.?     In  1  T.  and  2  cwt.  ? 

380.  Written  Exercises. 

1.  A  bushel  of  wheat  weighs  60  lb.  How  many 
bushels  of  wheat  will  weigh  IT.?   3J  T.  ?   5.4  T.  ? 

2.  When  wheat  is  worth  $.80  a  bushel,  how  much 
is  1  T.  worth  ? 

3.  When  wheat  is  selling  at  $1.75  per  hundred- 
weight, what  is  the  price  per  bushel  ? 

4.  A  farmer  sold  his  wheat  at  $18  a  ton.  He  liad 
63,896  lb.  of  wheat.  How  much  did  he  receive  for 
his  crop  ? 

5.  What  per  cent  of  a  pound  is  8  ounces  ? 


MEASURE  OF   LENGTH  249 

MEASURE   OF   LENGTH 

lyd. 

i_ 1 u I I I I 1 I I I I I I 1 

1  ft.  12  in. 

381.   1.    One  foot  is  what  part  of  1  yd.  ? 

2.  One  inch  is  what  part  of  1  ft.  ?     It  is  what  per 
cent  of  1  ft.  ? 

3.  What  part  of  1  ft.  are  4  in.  ?  6  in.  ?  9  in.  ?  8  in.  ? 
2  in.  ?   3  in.  ? 

4.  What  per  cent  of  1  ft.  are  4  in.  ?    6  in.  ?   9  in.  ? 
8  in.?   2  in.?    3  in.?    12  in.? 

5.  A  fathom  (6  ft.)  is  used  to  measure  the  depth 
of  the  sea. 

6.  A  chain  (4  rd.)  is  used  hy  surveyors  in  measur- 
ing land. 

7.  A  hand  (4  in.)  is  used  in  measuring  the  height 
of  a  horse. 

8.  A  horse  is  15  hands  high.     How  many  feet  high 
is  the  horse  ? 

SQUARE    MEASURE 

382.    See  p.  175. 

1.  Find  the  number  of  square  feet  and  the  number 
of  square  yards  in  a  surface  36  ft.  long  and  24  ft.  wide. 

2.  How  many  square  feet  are  there  in  the  floor  of 
your  schoolroom  ? 

3.  Find  the  number  of  square  yards  of  plastering 
there  are  in  your  schoolroom. 


250  DENOMINATE   NUMBERS 

CUBIC   MEASURE 

383.  1.  A  cubic  inch  is  a  solid  whose  six  equal 
sidea  are  each  1  sq.  in. 

2.  A  cubic  foot  is  a  solid  whose  six  equal  sides 
are  each  1  sq.  ft. 

3.  Examine  the  cubic  inch.  Has  it  length  ?  Has 
it  width  ?    Has  it  thickness  ? 

4.  How  many  cubic  inches  will  cover  a  square  foot 
of  surface  ?     Try  it. 

5.  If  you  make  them  2  deep,  how  many  cubic 
inches  can  you  place  on  a  square  foot  of  surface  ? 
3  deep?  12  deep?  The  pile  12  deep  will  be  a 
cubic  foot. 

7.  There  are  —  cubic  inches  in  a  cubic  foot. 

8.  Think  of  a  box  8  in.  long,  4  in.  wide,  and  3  in. 
high.  How  many  cubic  inches  could  be  placed  on  the 
bottom  of  the  box  ?  How  many  deep  could  you  place 
the  blocks  before  the  box  would  be  full  ? 

Model  : 
8  cu.  in.  =  the  number  of  cubic  inches  that  can  be 
placed  in  a  space  8  in.  long,  1  in.  wide, 
and  1  in.  high. 
32  cu.  in.  =  the  number  of  cubic  inches  that  can  be 
placed  in  a  space  8  in.  long,  4  in.  wide, 
and  1  in.  high. 
96  cu.  in.  =  the  number  of  cubic  inches  in  a  space  8  iiL 
long,  4  in.  wide,  and  3  in.  high. 


CUBIC  MEASURE  251 

384.    1.    Memorize  the  following  : 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd^) 

128  cubic  feet  =  1  cord  of  wood 

2.  Draw  a  cubic  inch.     Draw  a  cubic  foot. 

3.  How  can  you  find  how  many  cubic  inches  there 
are  in  2  cu.  ft.? 

4.  How  many  cubic  inches  are  there  in  3  cu.  ft.  ? 
In  3  cu.  ft.  and  120  cu.  in.? 

5.  A  man  dug  a  cellar  18  ft.  long,  12  ft.  wide,  and 
6  ft.  deep.  How  many  cubic  feet  of  earth  did  he 
remove  ?  How  many  cubic  yards  did  he  remove  ? 
He  was  paid  $.32  a  cubic  yard.  How  much  did  he 
receive  for  the  work? 

6.  A  trench  was  dug  3  ft.  6  in.  (3J  ft.)  wide  and 
12  ft.  deep,  in  which  to  lay  a  sewer.  The  sewer  w^as 
1  mile  (5,280  ft.)  long.  How  many  cubic  yards  of 
earth  were  removed  ? 

7.  Find  the  number  of  cubic  feet  in  a  space  12  ft. 
long,  8  ft.  wide,  and  6  ft.  deep. 

8.  Find  the  number  of  cubic  feet  in  your  school- 
room. 

9.  A  box  contains  60  cu.  ft.  It  is  5  ft.  long  and 
4  ft.  high.     How  wide  is  it  ? 

10.    A   room   contains   1620  cu.  ft.      It  is   15  ft. 
long  and  12  ft.  wide.     How  high  is  it? 


252  DENOMINATE  NUMBERS 

LUMBER  MEASURE 
385.    Lumber  is  measured  by  the  board  foot. 

A  board  foot  is  a  piece  of  lumber  one  foot  long,  one 
foot  wide,  and  one  inch  thick. 

7b  find  the  number  of  hoard  feat  in  a  piece  of 
lumber,  multiply  the  length  of  the  piece  in  feet  by 
the  thickness  of  the- piece  in  inches,  and  this  by  the 
tvidth  of  the  piece  in  inches,  and  divide  the  product 
by  12. 

To  shorten  the  work  use  cancellation. 

1.  Find  the  number  of  feet  of  lumber  in  a  piece 
of  lumber  16  it.  long,  9  in.  wide,  and  3  in.  thick. 


4 
Model  :    J^  x9x? 


36,  the  number  of  board  feet. 


2.  Find  the  number  of  feet  of  lumber  in  12  pieces, 
each  16  ft.  long,  8  in.  wide,  and  1  in.  thick. 

3.  Find  the  number  of  board  feet  in  a  timber  20  ft. 
long,  8  in.  wide,  and  8  in.  thick. 

4.  Find  the  number  of  board  feet  of  flooring  in  the 
floor  of  your  schoolroom. 

5.  At  $  12  per  thousand  feet,  what  will  be  the 
cost  of  20  pieces,  each  10  ft.  long,  4  in.  wide,  and 
2  in.  thick  ? 

6.  Find  the  cost  of  lumber  at  a  neighboring  lumber 
yard. 


CASH   ACCOUNT 


CASH  ACCOUNT 

386.    A  cash  account  is  a  written  statement  of  cash 
received  and  cash  paid  out. 

Dr.         Keceived  cash.  Paid  out.  Cr, 


1905 

1905 

Jan. 

1 

On  hand 

8 

75 

Jan. 

2 

By  cash 

4 

00 

a 

8 

To  salary 

12 

00 

a 

13 

"    board 

7 

00 

(( 

16 

To  remit 

30 

00  i 

a 

16 

"    bank  deposit 

30 

00 

11 

24 

To  salary 

12 
62 

00 1 

75 

a 

25 

Balance 

21 
62 

1^ 

Jan. 

25 

On  hand 

21 

75 

387.  1.  The  left-hand  side,  or  debit  side,  of  a  cash 
account  shows  the  cash  received  and  from  what  sources 
it  was  received. 

2.  What  does  the  right-hand,  or  credit  side,  of  a 
cash  account  show  ? 

3.  Why  should  one  keep  a  cash  account  ? 

4.  What  is  meant  by  the  entry  "  On  hand  $8.75  "  ? 

5.  The  above  was  Mr.  A's  cash  account  from 
January  1  to  January  25,  1905.  How  much  cash 
did  Mr.  A  receive  during  this  time  ? 

6.  With  the  $8.75,  how  much  cash  must  be  ac- 
counted for  ? 

7.  What  did  Mr.  A  do  with  his  money  ? 

a    How  much  money  had  Mr.  A  on  hand  Jan.  25  ? 
9.    What  is  meant  by  *'  Balance  "  ? 


254  DENOMINATE  NUMBERS 

10.  This  was  Harry's  cash  business  for  the  month  of 
February,  1904.  Rule  your  paper,  make  up,  and  close 
Harry's  account.  Feb.  1  Harry  haoy on  hand  $.45; 
Feb.  2  he  paid  out  for  papers  $.35,  ^d  received  for 
papers  $.70  ;  Feb.  3  he  received  for  weeding  a  garden 
("for  labor")  $.60;  Feb.  4  he  paid  for  papers  $.60, 
and  received  for  papers  $1.2(rpFeb.  8  he^~-paid 
$.10  carfare,  and  received  for  delivering  a  package 
$.35;  Feb.  9  he  bought  a  book  for  his  sister,  pay- 
ing $.20;  Feb.  12  he  earned  $1,  and  spent  for  car- 
fare $.20. 

11.  Make  similar  accounts. 

ANGLES 

388.  An  angle  is  the  opening  between  two  lines 
that  meet. 


Angle  Right  Angle  Acute  Angle  Obtuse  Angle 

1.  Join  two  lines  at  a  point  not  at  the  ends  of  the 
lines.  How  many  right  angles  is  it 
possible  to  make  with  two  lines  thus 
joined?  How  many  obtuse  angles  ? 
How  many  acute  angles  ? 

2.  A  right  angle  is  an  angle  formed  by  the  meet- 
ing of  one  straight  line  perpendicular  to  another. 

3.  An  acute  angle  is  an  angle  that  is  less  than  a 
right  angle. 


ANGLES  255 

4.  An  obtuse  angle  is  an  angle  that  is  greater  than 
a  right  angle. 

5.  How  many  angles  has  a  square  ?  An  oblong  ? 
What  kind  of  angles  are  they  ? 

6.  Draw  a  circle  on  the  blackboard.  Divide  it 
into  fourths.  How  many  right  angles  are  there  in 
the  circle? 

7.  In  the  figures  on  page  64,  which  angles  are 
right  angles  ? 

8.  The  line  that  bounds  a  circle  is  called  its 
circumference. 

9.  Angles  are  nieasured  in  degrees.  The  angles 
of  a  circle  are  measured  on  its  circumference.  There 
are  360  degrees  in  a  circle. 

10.  How  many  degrees  are  there  in  a  right  angle  ? 
In  one  half  of  a  right  angle  ? 

11.  Divide  a  right  angle  into  three  equal  angles. 
How  many  degrees  are  there  in  each  of  these  angles  ? 

12.  An  angle  of  -180  degrees  is  equal  to  two  right 
angles.  Explain  why  there  are  180  degrees  from  the 
north  pole  to  the  south  pole. 

13.  Explain  the  use  of  meridians  and  parallels. 

389.  CIRCULAR   MEASURE 

60  seconds  ('')  =  1  minute  (') 
60  minutes       =  1  degree  (°) 
360  degrees        =  1  circle 


256  ROMAN   NOTATION 


390. 

ROMAN 

NOTATION 

1    I 

10 

X 

lOO 

C 

1000   M 

2   II 

20 

XX 

200 

CC 

2000   MM 

3   III 

30 

XXX 

300 

CCC 

3000   MMM 

4    IV 

40 

XL 

400 

CD 

4000   IV 

5   V 

50 

L 

500 

D 

5000  V 

6   VI 

60 

LX 

000 

DC 

6000   VI 

7   VII 

70 

LXX 

700 

DCC 

7000   VII 

8   VIII 

80 

LXXX 

800 

DCCC 

8000   VIII 

9   IX 

90 

xc 

900 

CM 

9000   IX 

391.  1.  The  letters  used  in  Roman  notation  are : 
I,  V,  X,  L,  C,  D,  M. 

2.  When  a  letter  of  less  value  is  written  before  a 
letter  of  greater  value,  its  value  is  taken  from  that  of 
the  letter  of  greater  value ;  as,  IV,  IX,  XL.  When 
it  is. placed  after  a  letter  of  greater  value,  it  is  added; 
as,  VI,  XI,  LV,  etc. 

3.  A  dash  placed  over  a  letter  or  a  combination  of 
letters  increases  the  value  a  thousand-fold. 

4.  Write  in  Roman  notation :  27,  34,  68,  89,  235, 
309,  540,  894,  1000. 

5.  Write  in  Roman  notation:  1890,  1776,  1904, 
2000. 

6.  Write  in  figures  : 

CDXXI  MXCIV  MDCCCXCIX 

7.  Where  have  you  seen  the  Roman  numerals  used  ? 


YB  35856 


ivil8750J2 


■=':> 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


